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+{"text":"\\section{Introduction}\nOne of the elemental forms of document processing includes classification. \nSince the last couple of years, it is in demand because of the increasing \navailability of data in digital format which has resulted into the  requirement of systematization of  \nthat data. Manual organization of huge data can be tedious if strict time \nconstraints are set, increasing the necessity of automated document classification. The contexts of words in the documents play a very important role in deciding the category of the document. The human brain is very effective in consideration of contexts in the incoming information for taking the appropriate action. \n\n\\smallskip\n\nThe principles of HTM theory can be used to meet the requirements of organizing of data.  \nHTM \ntakes inspiration from the mammalian brain which has been evolving over \nmillions of years and is able to process data efficiently. As HTM is \nbiologically plausible, it is based on simple rules and not complex mathematics. HTM theory is being developed by a US based company called Numenta, Inc.\n\\smallskip\n\n\\section{Related Work}\nSome of the conventional methods for text\/document classification are mentioned below:\n\\subsection{Naive Bayes}\nThe Naive Bayes classifier is a probabilistic classifier and is based on the Bayes theorem. It works well with small samples of data. The posterior probability of a particular document belonging to various classes is calculated. The document is assigned to the class with the highest posterior probability. The Naive Bayes classifier assumes strong independence between the features. This is a major limitation of this classifier and hence has low performance in cases where the features are correlated \\cite{comparison}.\n\n\\subsection{Support Vector Machines}\nSupport Vector Machines (SVMs) are supervised machine\nlearning algorithms. In case of a multi class problem, first\nthe problem has to be decomposed into two separate class\nproblems as SVM can work only with binary classification\nproblem. They will probably give poor results when total\nnumber of samples are very less than the total number of\nfeatures. In comparison with decision making classifier and\nlogistic regression, SVM takes more time for computation \\cite{comparison}.\n\n\\subsection{K-Nearest Neighbour}\nK-Nearest Neighbour (KNN) is used for classification of objects by calculating the distance of training samples from each object. KNN classification is a simple and widely used approach for text classification. However, it is computationally intensive and\nclassification time is high \\cite{comparison}. Also, it is difficult to find the ideal value of k \\cite{knn}.\n\n\n\\subsection{Convolutional Neural Network}\nConvolutional Neural Network (CNN) works well with static text classifications. CNN is a type of feed forward neural network, comprising of neurons with trainable weights and biases. CNN comprises of a number of convolutional layers with nonlinear activation functions like ReLU or tanh applied to the results. CNN suffers from the limitations of the requirement of large data and big processing power to be able to predict accurately \\cite{cnn}.\n\nHTM theory is primarily used for Classification, Prediction and Anomaly Detection purposes. One of its application for Classification is mentioned below:\n\n\\subsection{Land - forms classification}\nAs HTM based models have a common learning algorithm, it can be used for classifying images. HTM theory has been used for classifying different land-forms like trees, roads, buildings and farms using the images obtained from satellites. The framework used achieved an accuracy of 90.4\\%,\\cite{perea2009application}  which is at par with the conventional machine learning techniques for image classification.\n\nSince HTM theory can be used for image classification purposes, it can hold a promise to classify text\/documents.\n\n\\section{Overview of Hierarchical Temporal Memory}\nHTM is a theory which seeks to apply the \nstructural as well as algorithmic properties of the neocortex to machine \nlearning problems \\cite{hawkins2010hierarchical}. The neocortex proves to be the center of intelligence in the \nmammalian brain. It is responsible for processing complex activities such as \ncommunication, planning and prediction. Structurally, neocortex is a 2 mm thick \ntissue divided into a number of different regions. A region is a network of \ninterconnected neurons \\cite{hawkins2010hierarchical}. This attributes to the presence of input connections \nfrom different sensory organs \\cite{hawkins2006hierarchical,onintelligence} like eyes, ears etc. The term \"Hierarchical\" in the theory is owing to \nthe fact that, HTM network contains a hierarchy of levels arranged in the \npyramid-like structure. These levels are present in the form of regions that \nare again composed of columns which finally consist of neurons. These neurons need \nnot be physically arranged in a hierarchy, but are logically arranged in the \nhierarchical format. The lower levels in hierarchy represent data having lower \nabstraction\/complexity. As we go higher in the hierarchy, the data abstraction stored in the memory increases. Time plays a crucial role in the \nway data is stored in mammalian brain. \"Temporal\" implies that the HTM network takes \ninto consideration the sequence of the incoming data. A continuous stream of \ninput data is aptly learned as spatial and temporal sequences.\n\\smallskip\n\nA remarkable property of the neocortex is that the input from all the sensory \norgans is processed in the same manner. Hence, it has a common learning algorithm \nfor inputs from all types of sensory organs \\cite{hawkins2010hierarchical}. \n\n\\begin{figure*} [htb!] \n\\includegraphics[width=1\\textwidth]{process_htm_final.png}\n\\caption{\\sl System Architecture Diagram\n\\label{fig:implementation}}\n\\end{figure*} \n\n\n\n\\subsection{Structure of a Neuron}\n\n\nInside the mammalian brain, neurons play a central role in information handling. Some \nrelevant parts of the neuron for our study are mentioned below. \n\n\\subsubsection{Proximal Dendrites}\nProximal dendrites are in close proximity to the cell body. The proximal \ndendrites are connected directly to the inputs from the sensory \norgans. \n\n\n\\subsubsection{Distal Dendrites}\nDistal dendrites are the ones that are afar from the cell body. The distal \ndendrites have connections with various other neurons in the neocortex. Majority of the \nconnections to the axon are from distal dendrites as compared to the \nconnections made by proximal dendrites \\cite{synapse}. \n\n\n\\subsubsection{Synapse}\nA synapse is a connection between an axon of one neuron and dendrite of the \nother. The ongoing process of breaking and reforming these synapses between \ncells results in learning of new data and thus gradually forgetting the old \none.\\smallskip\n\nThere is a permanence value associated with every synapse and a threshold \nlinked with every neuron. Thus, for a neuron to get activated, the total number \nof synapses with permanence values higher than the threshold value must be more \nthan the stimulus threshold.\n\n\\subsection{Sparse Distributed Representation}\nThough the neocortex contains billions of neurons in highly interconnected \nmanner, only a tiny fraction of them are active for a particular input \\cite{sdr1}. Hence, only small percentage of active neurons are responsible for \nrepresenting the input information. This is called as Sparse Distributed Representation (SDR). \nEven though single activated neuron has the potential to convey some meaning, \nthe full information can only be conveyed when it is interpreted within the \ncontext of other neurons.\nAs the information is spread across a tiny percentage of the active bits, SDRs are more noise tolerant than dense representations, making them ideal for text processing.\n\n\\subsection{Spatial Pooler}\nHTM includes two important parts - Spatial Pooler (SP) and Temporal Pooler (TP). Spatial Pooler, also known as Pattern Memory, has been emphasized in this study.\n\\smallskip\n\nThe neurons in the neocortex are arranged in columns, which represent features of the input. Every neuron in a particular column, which represent different context for an input, is \nconnected to specified number of bits in the input bit array. The selection of \nbits to be connected to the neurons in a particular column is random. The bits \nwhich are connected to a particular column are known as a potential pool of \nthat particular column. Connections between input bit and the column neuron is \ncalled as a synapse. Every synapse has a value associated with it known as \npermanence value similar to that of a mammalian brain. Permanence value is always in the range of 0 and 1. There is a threshold value associated with synapse's permanence.\n\nIf the permanence value of a synapse associated to an input bit is greater than the threshold, the activation of the column of neurons \nis influenced by the input bit. The permanence value of a synapse is adjusted in the learning phase.\n\n\nThe main role of SP in HTM is finding spatial patterns in the input data. It is decomposed into three stages:\n\\medskip\n\\subsubsection{Overlap}\nIn this stage, overlap score of each column is calculated. Overlap score is \n the count of active bits in the potential pool of a particular \ncolumn having permanence value greater than the threshold. \n\\smallskip\n\\subsubsection{Inhibition}\nThe columns are sorted according to their overlap scores from highest to lowest. A particular fraction (in our study, 0.5\\%, Table \\ref{spatial_pooler_paramters}, $NumActiveColumnsPerInhArea$) of the top columns is selected (also called as active columns or the winning columns) for the learning phase. Rest other columns are inhibited from learning.\n\\smallskip\n\\subsubsection{Learning}\nDuring Learning, the permanence value of the synapses in the potential pool of the winning \ncolumns is incremented (by synPermActiveInc, Table \\ref{spatial_pooler_paramters}) or decremented (by synPermInactiveDec, Table \\ref{spatial_pooler_paramters}). When the active column is connected to an \nactive bit then the permanence value of the synapse corresponding to that \nactive bit is incremented. However, when the active column is connected to an\ninactive bit then the permanence value of the synapse corresponding to that \ninactive bit is decremented. This is the result of column expecting that bit to be active. The synapse permanence is decremented as a \npunishment. \n\n\\section{Implementation}\n\nThe flowchart in figure \\ref{fig:implementation} is our high-level \narchitecture diagram for document categorization. As the mammalian brain requires electrical \nsignals for learning, the learning algorithm i.e., Spatial Pooler also requires \nbit patterns for processing. So, to convert text into bit arrays, Latent Semantic \nIndexing (LSI) technique is used, which converts semantically similar sentences \ninto similar bit arrays. These bit arrays (which need not be sparse)\nare fed to the Spatial Pooler where it simulates the working of neurons in \nthe brain and gives SDR as the output. The \nactive bits in the SDR represent the neurons \nwhich get activated in the Spatial Pooler. Since semantically similar \ntext belong to the same category, it is easy to classify the \ntext into different categories.\n\\medskip\n\\subsection{Latent Semantic Indexing}\nAs HTM theory is modelled after the mammalian brain, its input also should be in \naccordance with the input format received by the brain. The brain receives \ninput in the form of electrical signals which correspond to bit arrays. Latent Semantic Indexing(LSI) helps in determining hidden \nfeatures in documents \\cite{papadimitriou1998latent}. Thus the technique is used to extract the \ncontextual-usage meaning of words from the documents\\cite{lsa3}. \nThe LSI framework consists of 3 steps which are mentioned below.\n\\smallskip\n\n\n\\subsubsection{Preprocessing of input data}\nIn the initial step, the input text is tokenized and stopwords are removed from \nevery document of the corpus\\footnote{We have used the wikipedia language corpus as it \nincludes a large vocabulary which is useful for generic datasets. The corpus \nwill change if the dataset is in a language other than English or contains a \nlarge number of words which are not present in the vocabulary.}. Each term in \nthe text is then represented as a tuple containing term-id and term frequency. \nA matrix is created in which the rows denote the unique terms and the \ncolumns denote the documents. Every cell denotes the term count in the corresponding document.\nThe matrix of term-frequency counts obtained from the term document matrix is \nthen modified using the TF-IDF technique so as to give more weight to rare \nterms compared to common terms across documents and also to frequently occurring \nterms in a particular document.   \nThe formula for weighing each term can be represented as,\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=1\\linewidth]{LSA_with_bold_new.png}\n\\caption{\\sl Latent Semantic Indexing framework\n\\label{fig:lsa_framework}}\n\\end{centering}\n\\end{figure} \n\n\n\\begin{gather}\n  Document Term Weight = f_{t,d} \\times \\ln(N \/ n_t) \\\\\n\\intertext{Where:}\n  \\begin{tabular}{>{$}r<{$}@{\\ :\\ }l}\n    f_{t,d} & count of term~$t$ in document~$d$ \\\\       \n    N & the total count of documents \\\\\n    n_t & the count of documents having term~$t$ \\\\\n  \\end{tabular}\\nonumber\n\\end{gather}\n\n\\smallskip\n\n\nThe term-document matrix gets modified to contain weights of each term \nin a given document. The dimensionality reduction of this matrix is done using \nSingular Value Decomposition (SVD).\n\\smallskip\n\\subsubsection{Singular Value Decomposition}\nLSA uses SVD for generating the vectors of a particular text \\cite{lsa1, lsa2}. The matrix $X$(term-document) is used to calculate two matrices. These are,\n\n\\begin{gather}\n  Y = X^TX \\\\\n  Z = XX^T \\\\\n\\intertext{Where:}\n  \\begin{tabular}{>{$}r<{$}@{\\ :\\ }l}\n    $X$ & term - document matrix \\\\\n\t$Y$ & document - document matrix \\\\       \n    $Z$ & term - term matrix \\\\\n  \\end{tabular}\\nonumber\n\\end{gather}\n\nAfter finding eigenvectors of $Y$ and $Z$ matrices, we get left singular matrix, $L$ \nand right singular matrix, $R$ respectively. Thus, term - document matrix, $X$, \n is divided into unique combination of three matrices as follows -\n\n\\eat{\n\\begin{gather}\n X = L \\Sigma R^T \\\\\n\\intertext{Where:}\n  \\begin{tabular}{>{$}r<{$}@{\\ :\\ }l} \n  \\multirow{2}{*}{$L$} & is the left singular vector matrix representing \\\\\n  & weights of a term for corresponding concepts. \\\\\n  \\multirow{2}{*}{$R^T$} & is the transpose of the right singular vector matrix \n\\\\\n  & representing  weights of documents belonging to particular concepts. \\\\\n\\smallskip\n\\multirow{2}{*}{$\\Sigma$} & is the diagonal matrix representing \\\\\n  & weights of the concepts found in the text\\\\\n  \\end{tabular}\\nonumber\n\\end{gather}\n}\n\n\\begin{gather}\n\\label{4}\n X = L \\Sigma R^T \\\\\n\\intertext{Where:}\n  \\begin{tabular}{>{$}r<{$}@{\\ :\\ }l} \n  $L$ & Term - Concept weight matrix \\\\\n  R^T & Concept - Document weight matrix \\\\\n  \\Sigma & Diagonal matrix representing concept weights \\\\\n  \\end{tabular}\\nonumber\n\\end{gather}\n\n$\\Sigma$ is calculated by taking the square root of the eigenvalues of matrix $Y$. \n\\smallskip\n\nTo reduce the dimensionality of the matrices in equation \\ref{4}, top $k$ concepts are selected and thus \nmatrix $X$ is approximated as,\n\n\\begin{gather}\n X_k = L_k \\Sigma_k R_k^T\n\\end{gather}\n\n\nIn our study, $k$ is taken to be 400 in order to consider top 400 concepts. This \nmarks the end of the training phase. \n\nIn the testing phase, after generating \nweight matrix using the Term Frequency - Inverse Document Frequency (TF - IDF) \nmodel, input text gets converted into a query matrix, $Q$. This matrix $Q$ is then \nmultiplied with matrices $L_K$ and $\\Sigma_k$ to generate new query vectors calculated as follows:\n\n\\begin{gather}\n  New Query Vectors = Q L_k \\Sigma_k\n\\end{gather}\n\n\\subsubsection{Extraction of top features}\n\nThe query vectors are converted into bit arrays of size 400. \nThe indices of the top 40 features from the query vectors represent the '1's in the bit arrays and the indices of the remaining features represent '0's.\n\n\\subsection{Spatial Pooler}\nThe bit arrays from the LSI encoder are then passed to the Spatial Pooler for \nlearning. The Spatial Pooler gives similar Sparse Distributed Representations \n(SDRs) for similar input text. The major parameters of the Spatial Pooler which \nsignificantly affect the accuracy of our model are mentioned in Table \\ref{spatial_pooler_paramters}.\n\n\\medskip\n\n\\begin{table}[htbp]\n\\caption{Spatial Pooler Parameters}\n\\label{spatial_pooler_paramters}\n\\begin{center}\n\n\\begin{tabular}{|l|c|}\n\t\\hline\n\t\\textbf{Parameters} & \\textbf{Values}\\\\\n\t\\hline\n    inputDimensions & 400\\\\\n\t\\hline\n    columnDimensions & 20000\\\\\n\t\\hline\n    potentialRadius & 200\\\\\n\t\\hline\n    numActiveColumnsPerInhArea & 100\\\\\n\t\\hline\n    synPermActiveInc & 0.01\\\\\n\t\\hline\n    synPermInactiveDec & 0.008\\\\\n\t\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\smallskip\nThe active indices of the SDR are then fed to the Classifier.\n\n\\subsection{Classifier}\nIn order to predict target class labels, a sequence of N-dimensional SDRs is \nassigned to a set of k class labels. The Classifier makes \nuse of a single layer feed forward neural network. In the figure \n\\ref{fig:classifier}, the number of output neurons is equal to the number of \npredefined categories. The number of input neurons is equal to the number of \nbits in any SDR.\n\n\\begin{figure}  \n\\includegraphics[width=0.9\\linewidth]{sdr_classifier.png}\n\\caption{\\sl Classifier\n\\label{fig:classifier}}\n\\end{figure} \n\nThe algorithmic description of the classifier is as follows,\n\\smallskip\n\\subsubsection{Matrix Initialisation}\nSince all classes have an equal chance of occurrence before learning, all \nvalues in the weight matrix are initialised to zero. \n\\smallskip\n\\subsubsection{Inference}\nIn this phase, the predicted class probabilities for each input pattern are \ncalculated. The calculations include two steps as mentioned below.\n\\smallskip\n\\setlength{\\parindent}{4ex}\n\n\\textit{i) Weighted sum :} Weighted sum of the input is calculated for each output neuron to determine the activation levels of the neuron. \nActivation level of an output neuron can be determined by the summation of the product of all the input bits to the input layer neurons with the weights of its corresponding connections to the output layer neuron. \nThe formula for the activation level being,\n\n\n\\begin{gather}\n a_j = \\sum_{i=1}^{N} w_{ij} \\times x_i \\\\\n\\intertext{Where:}\n  \\begin{tabular}{>{$}r<{$}@{\\ :\\ }l} \n  a_j & activation level of the $j^{th}$ output layer neuron \\\\\n  n & number of input layer neurons\\\\\n  \\eat{w_{ij} & weight of the connection from the $i^{th}$ input layer  \n\\\\ & neuron to the $j^{th}$ output layer neuron \\\\}  \n   \\multirow{2}{*}{$w_{ij}$} & weight of the connection from the $i^{th}$ input \nneuron \\\\\n  & to the $j^{th}$ output neuron. \\\\\n  x_i & Input bit value. It is either 0 or 1. \\\\\n  \\end{tabular}\\nonumber\n\\end{gather}\n\n\\par\n\n\\setlength{\\parindent}{4ex}\n\n\\textit{ii) Softmaxing the activation levels :} The probability distribution of \nthe categories is calculated by exponentiating and normalizing the activation \nlevels of the neurons in the output array using the softmax function. The \nformula for the probability distribution being,\n\n\\begin{gather}\n P\\left[ C_k | x, w\\right] = y_k = \\frac{e^{a_k}}{\\sum_{i=1}^{k} e^{a_i}} \\\\\n\\intertext{Where:}\n  \\begin{tabular}{>{$}r<{$}@{\\ :\\ }l} \n  y_k & Probability of predicting the category index $k$. \\\\\n  k & Number of predefined categories. \\\\\n  \\end{tabular}\\nonumber\n\\end{gather}\n\n\\par\n\n\\subsubsection{Learning}\nDuring each iteration, the classifier makes a prediction of the category index \nof a given SDR. This prediction is of the form of the probability distribution \nover different category indexes.\nThe connection weights are updated to learn and improve the prediction results. Connection weights are adjusted only for the active bits. \nThe Connection Weights are determined using maximum likelihood estimation (MLE) on independent \ninput SDRs. Since, the SDRs are independent of each other, they would satisfy \nthe following equation.\n\n\\begin{gather}\n P\\left[ z^1, z^2, ..., z^t\\right] =  \\prod_{t} P\\left( z^t | x^t, w\\right) \\\\\n x^t = (x^{t}_1, x^{t}_2, x^{t}_3, ..., x^{t}_N) \\\\\n\\intertext{Where:}\n  \\begin{tabular}{>{$}r<{$}@{\\ :\\ }l} \n  z^t & actual category index of $t^{th}$ SDR. \\\\\n  x^t & Sparse Distributed Representation. \\\\\n  x^{t}_1 & first bit of the $t^{th}$ SDR. \\\\\n  w & Connection weights. \\\\\n  \\end{tabular}\\nonumber\n\\end{gather}\n\nA value of $w$ is selected so that likelihood gets maximized. The loss function to select $w$ \\cite{neuneier2012train}, is as follows : \n\n\\begin{gather}\n L = -\\ln \\left(  \\prod_{t} p\\left(y_t | x_t, w\\right)\\right) \\\\\n= -\\Sigma_{t}\\ln P\\left(y_t | x_t, w\\right)\n\\end{gather}\n\nGradient descent is used is to minimize the loss function.\n\n\\begin{gather}\n\\frac{\\partial L}{\\partial w_{ij}} = \\frac{\\partial L}{\\partial a_j} \\times \n\\frac{\\partial a_j}{\\partial w_{ij}} \\\\\n=\\left(y_j - z_j\\right)x_i \\\\\n\\intertext{Where:}\n  \\begin{tabular}{>{$}r<{$}@{\\ :\\ }l} \n  y_j & Predicted probability of $j^{th}$ category index. \\\\\n  z_j & Actual probability of $j^{th}$ category index. \\\\\n  x_i & Input bit value to the $i^{th}$ input neuron. \\\\\n  \\end{tabular}\\nonumber\n\\end{gather}\n\nError in connection weight between $i^{th}$ input neuron to the $j^{th}$ output neuron is,\n\n\\begin{gather}\nError_{ij}\\ for\\ active\\ input\\ bits = y_j - z_j\n\\end{gather}\n\n\\begin{gather}\nupdate_{ij} = \\alpha \\times error_{ij} \\\\\n\\intertext{Where:}\n  \\begin{tabular}{>{$}r<{$}@{\\ :\\ }l} \n  \\alpha & Learning rate. \\\\\n  \\end{tabular}\\nonumber\n\\end{gather}\n\nThe value $update_{ij}$ is used to update the connection weight between the \n$i^{th}$ input neuron to the $j^{th}$ output neuron using the formula,\n\n\\begin{gather}\nw_{new_{ij}} = w_{old_{ij}} + update_{ij}\n\\eat{\\intertext{Where:}\n  \\begin{tabular}{>{$}r<{$}@{\\ :\\ }l} \n  \\alpha & Learning rate. \\\\\n  \\end{tabular}\\nonumber}\n\\end{gather}\n\nBut, if we just want to update connection weights for the bits which are active \nwe multiply $update_{ij}$ with $x_i$.\n\n\\begin{gather}\nw_{new_{ij}} = w_{old_{ij}} + update_{ij} \\times x_i\n\\intertext{Where:}\n  \\begin{tabular}{>{$}r<{$}@{\\ :\\ }l} \n  w_{new_{ij}} & updated weight of the connection from $i^{th}$ input \\\\\n  &  neuron to the $j^{th}$ output neuron \\\\\n  \\end{tabular}\\nonumber\n\\end{gather}\n\nThe output layer neuron with the highest probability represents the category index of \nthe input text.\n\n\\section{Results}\nMany experiments were performed to test the accuracy and performance of \nour model. We selected two standard datasets for document classification,\nnamely, 20 Newsgroup dataset from the sklearn dataset repository and Movie \nReviews dataset from the NLTK corpus repository. The datasets were \nsplit into train set and test set in the ratio 9:1. The classification framework used in this study gives comparable accuracies with the models mentioned in the table \\ref{tp_rate} on the same datasets. \n\\smallskip\n\n\\begin{table}[htbp]\n\\caption{TRUE POSITIVE RATE}\n\\label{tp_rate}\n\\begin{center}\n\n\\begin{tabular}{ | m{6.5em} | m{2cm}| m{2cm} | } \n\n\\hline\n\\bfseries Classification Techniques & \\bfseries 20 newsgroup & \\bfseries Movie \nReviews \\\\ \n\\hline\nSVM\\cite{svmmr} & ---- & 84.40\\% \\\\ \n\\hline\nDecision Trees\\cite{decisiontreesmr} & ---- & 61.10\\% \\\\ \n\\hline\nNaive Bayes\\cite{naivebayes20,naivebayesmr} & 86.00\\% & 62.35\\% \\\\ \n\\hline\nB-Tree\\cite{btree20} & 82.64\\% & ---- \\\\ \n\\hline\nBayesian Networks \\cite{bayes_net}& 78.58\\% & ---- \\\\\n\\hline\nHTM & 83.19\\% & 73.60\\% \\\\\n\\hline\n\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\smallskip\n\n\\section{Conclusion and Future Scope}\n\nThis paper puts forward the results of using the Hierarchical Temporary Memory \nmodel for document categorization. The results prove that the HTM model gives \nan accuracy comparable to the conventional techniques used for text \nclassification. \nThe number of columns and the SDR sparsity has a significant effect on the \nperformance of the spatial pooler. As per our model, The optimal values of the \nnumber of columns was 20,000 and the sparsity was 0.5\\%.\n\nThe main advantages of this model are: a limited number of parameters, can be \ntrained on small\\cite{synapse} corpus and faster training.\n\nIn future, we plan to modify the encoding process of our model and also \nincorporate the Temporal Pooler which can help to increase the accuracy of the \nmodel.\n\n\\section{Acknowledgement}\nWe are grateful to Mr. Nikhil Malhotra of Maker's Lab, Tech Mahindra Ltd. and Mr. Satish Kumbhar of College of Engineering, Pune, for guiding us through the research.\n\n\\nocite{*}\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction\\label{s.1}}\n\n\nThe construction of diffusions on infinite-dimensional manifolds and the study of the regularity properties of their induced measures have been a topic of great interest for at least the past 50 years; see for example \\cite{DalettskiSnaiderman1969,Kuo1971,Kuo1972,Elworthy1975,FdlP1995,ADK2003,AM2006,Malliavin08}, although many other references exist.  The purpose of the present paper is to construct diffusions on a general class of infinite-dimensional nilpotent Lie groups, and to show that the associated heat kernel measures are quasi-invariant under  appropriate translations.  We demonstrate that this class of groups is quite rich. We focus here on the elliptic setting, but comment that, as nilpotent groups are standard first models for studying hypoellipticity, examples of infinite-dimensional versions of such spaces are important for the more general study of hypoellipticity in infinite dimensions.  This is an area of great interest, and is of particular relevance in the study of stochastic PDEs and their applications \\cite{AiraultMalliavin2004,BakhtinMattingly2007,BaudoinTeichmann2005,HairerMattingly2011,Malliavin1990,MattinglyPardoux2006}. The present paper studies heat kernel measures for elliptic diffusions on these spaces, which is a necessary precursor to understanding the degenerate case.\n\n\n\\subsection{Main results}\nLet $(\\mathfrak{g},\\mathfrak{g}_{CM},\\mu)$ denote an abstract Wiener\nspace, where $\\mathfrak{g}$ is a Banach space equipped with a\ncentered non-degenerate Gaussian measure $\\mu$ and associated\nCameron--Martin Hilbert space $\\mathfrak{g}_{CM}$.  We will assume\nthat $\\mathfrak{g}_{CM}$ additionally carries a nilpotent Lie algebra\nstructure $[\\cdot,\\cdot]: \\mathfrak{g}_{CM}\\times \\mathfrak{g}_{CM}\n\\rightarrow \\mathfrak{g}_{CM}$, and we will further assume that\nthis Lie bracket is Hilbert-Schmidt.  We will call such a space an {\\it\nabstract nilpotent Lie algebra}. Via the standard\nBaker-Campbell-Hausdorff-Dynkin formula, we may then equip\n$\\mathfrak{g}_{CM}$ with an explicit group operation under which\n$\\mathfrak{g}_{CM}$ becomes an infinite-dimensional group.  When\nthought of as a group, we will denote this space by $G_{CM}$.  \nWe equip $G_{CM}$ with the left-invariant Riemannian metric which\nagrees with the inner product on $\\mathfrak{g}_{CM}\\cong T_eG_{CM}$, and we denote the\nRiemannian distance by $d$.  \nNote that, despite the\nuse of the notation $\\mathfrak{g}$, we do not assume that the Lie bracket\nstructure extends to $\\mathfrak{g}$, and so this space is not necessarily a\nLie algebra or a Lie group.  Still, when $\\mathfrak{g}$ is playing a role\ntypically played by the group, we will denote $\\mathfrak{g}$ by $G$.\n\nIf $\\mathfrak{g}$ were a Lie algebra (and thus carried an associated group\nstructure), we could construct a Brownian motion on\n$G$ as the solution to the Stratonovich stochastic differential equation\n\\[ \\delta g_t = g_t\\delta B_t := L_{g_t*}\\delta B_t, \\text{ with }\n\tg_0=\\mathbf{e}=(0,0), \\]\nwhere $L_x$ is left translation by $x\\in G$ and $\\{B_t\\}_{t\\ge0}$ is a\nstandard Brownian motion on $\\mathfrak{g}$ (as a Banach space) with $\\mathrm{Law}(B_1)=\\mu$.\nIn finite dimensions, the solution to this stochastic differential equation may be obtained\nexplicitly as a formula involving the Lie bracket.  In particular, for $t>0$\nand $n\\in\\mathbb{N}$, let $\\Delta_n(t)$ denote the simplex in\n$\\mathbb{R}^n$ given by\n\\[ \\{s=(s_1,\\cdots,s_n)\\in\\mathbb{R}^n: 0<s_1<s_2<\\cdots<s_n<t\\}. \\]\nLet $\\mathcal{S}_n$ denote the permutation group on $(1,\\cdots,n)$,\nand, for each $\\sigma\\in \\mathcal{S}_n$, let $e(\\sigma)$ denote the\nnumber of ``errors'' in the ordering\n$(\\sigma(1),\\sigma(2),\\cdots,\\sigma(n))$, that is, $e(\\sigma)=\\#\n\\{j<n: \\sigma(j)>\\sigma(j+1)\\}$.\nThen the Brownian motion on $G$ could be written as \n\\begin{equation}\n\\label{e.ibm}\ng_t = \\sum_{n=1}^{r-1} \\sum_{\\sigma\\in\\mathcal{S}_n}  \n\t\\left( (-1)^{e(\\sigma)}\\bigg\/ n^2 \n\t\\begin{bmatrix} n-1 \\\\ e(\\sigma) \\end{bmatrix}\\right) \n\t\\int_{\\Delta_n(t)} \n\t[ [\\cdots[\\delta B_{s_{\\sigma(1)}},\\delta B_{s_{\\sigma(2)}}],\\cdots], \n\t\\delta B_{s_{\\sigma(n)}}],\n\\end{equation}\nwhere this sum is finite under the assumed nilpotence.  \nAn obstacle to the development of a general theory of stochastic differential equations on infinite-dimensional Banach spaces is the lack of smoothness of the norm in a general Banach space which is necessary to define a stochastic integral on it.\nStill, in Section \\ref{s.mult}, we prove a general result to define a class of iterated stochastic integrals with respect to Brownian motion on the Banach space $\\frak{g}$ that includes the expression above.  Additionally, we show that one may make\nsense of the above expression when the Lie bracket on $\\mathfrak{g}_{CM}$ does not necessarily extend to $\\mathfrak{g}$. Thus we are able to define a ``group Brownian motion'' $\\{g_t\\}_{t\\ge0}$ on $G$ via (\\ref{e.ibm}). We let $\\nu_t:=\\mathrm{Law}(g_t)$ be the heat kernel measure on $G$.\n\nIn particular, the integrals above are defined as a limit of stochastic integrals on finite-dimensional subgroups $G_\\pi$ of $G_{CM}$. We show that these $G_\\pi$ are nice  in the sense that they approximate $G_{CM}$ and that there exists a uniform lower bound on their Ricci curvatures.  \n\nUsing these results, we are able to prove the following main theorem.  \n\n\n\\begin{thm}\n\\label{t.quasi}\nFor $h\\in G_{CM}$, let $L_h,R_h:G_{CM}\\rightarrow G_{CM}$ denote left and\nright translation by $h$, respectively. Then $L_h$ and $R_h$ define measurable\ntransformations on $G$, and for all $T>0$, $\\nu_t\\circ L_h^{-1}$ and $\\nu_t\\circ\nR_h^{-1}$ are absolutely continuous with respect to $\\nu_t$.  Let \n\\[ J_t^l(h,\\cdot) := \\frac{d(\\nu_t\\circ L_h^{-1})}{d\\nu_t} \\qquad \\text{ and }\n\t\\qquad J_t^r(h,\\cdot) := \\frac{d(\\nu_t\\circ R_h^{-1})}{d\\nu_t} \\]\nbe the Radon-Nikodym derivatives, $k$ be the uniform lower bound on the Ricci\ncurvatures of the finite-dimensional approximation groups $G_\\pi$ \nand \n\\[ c(t) := \\frac{t}{e^t-1}, \\qquad \\text{ for all } t\\in\\mathbb{R}, \\]\nwith the convention that $c(0)=1$.  Then, for all $p\\in[1,\\infty)$,\n$J_t^l(h,\\cdot),J_t^r(h,\\cdot)\\in L^p(\\nu_t)$ and both satisfy the estimate\n\\[ \\|J_t^*(h,\\cdot)\\|_{L^p(\\nu_t)} \\le \\exp \\left( \\frac{c(kt)(p-1)}{2t}\n\td(\\mathbf{e},h)^2\\right), \\]\nwhere $*=l$ or $*=r$.\n\\end{thm}\n\nThe fact that one may define a measurable left or right action on $G$ by an\nelement of $G_{CM}$ is discussed in Section \\ref{s.mga}.  The lower bound on\nthe Ricci curvature is proved in Proposition \\ref{p.Ric}.\n\n\n\\subsection{Discussion}\n\nThe present paper builds on the previous work in \\cite{DriverGordina2008} and\n\\cite{Melcher2009}, significantly generalizing\nthese previous works in several ways.\nIn particular, the paper \\cite{Melcher2009} considered\nanalogous results for ``semi-infinite Lie groups'', which are \ninfinite-dimensional nilpotent Lie groups\nconstructed as extensions of finite-dimensional nilpotent Lie groups\n$\\mathfrak{v}$ by an\ninfinite-dimensional abstract Wiener space (see Example \\ref{ex.ext}).  \nAt several points in the analysis\nthere, the fact that $\\mathrm{dim}(\\mathfrak{v})<\\infty$ was used in a\ncritical way.  In particular, it was used to show that the stochastic\nintegrals defining the Brownian motion on $G$ as in equation\n(\\ref{e.ibm}) were well-defined.  \nIn the present paper, we have removed this restriction, as well as removing the\n``stratified'' structure implicit in the construction as a Lie group extension.\n\nAgain, we note that, despite the\nuse of the notation $\\mathfrak{g}$, it is not assumed that the Lie bracket\nstructure on $\\mathfrak{g}_{CM}$ extends to $\\mathfrak{g}$, and so $\\mathfrak{g}$ itself is not necessarily a\nLie algebra or Lie group.  In \\cite{DriverGordina2008} and \\cite{Melcher2009}, it\nwas assumed that the Lie bracket was a continuous map defined on\n$\\mathfrak{g}$.  However, it turns out that\nthe group construction on $\\mathfrak{g}_{CM}$ is  the only\nnecessary structure for the subsequent analysis.\nAs is usual for the infinite-dimensional setting,\nwhile the heat kernel measure is itself supported on the larger space\n$\\mathfrak{g}$, its critical analysis depends more on the structure of $\\mathfrak{g}_{CM}$.\nStill, as was originally done in \\cite{DriverGordina2008} and then in \\cite{Melcher2009},\none may instead define an abstract nilpotent Lie algebra starting \nwith a continuous nilpotent bracket\n$[\\cdot,\\cdot]:\\mathfrak{g}\\times\\mathfrak{g}\\rightarrow\\mathfrak{g}$.  For\nexample, in the event that $\\mathfrak{g}=W\\times\\mathfrak{v}$ where\n$\\mathfrak{v}$ is a finite-dimensional Lie algebra and\n$[\\cdot,\\cdot]:\\mathfrak{g}\\times\\mathfrak{g}\\rightarrow\\mathfrak{v}$, it is\nwell-known that this implies that the restriction of the bracket to\n$\\mathfrak{g}_{CM}=H\\times\\mathfrak{v}$ is Hilbert-Schmidt.  (For\nany continuous bilinear $\\omega:W\\times W\\rightarrow K$ where $K$ is a Hilbert\nspace, one has that $\\|\\omega\\|_{(H^{\\otimes 2})^*\\otimes K}<\\infty$; this\nfollows for example from Corollary 4.4 of \\cite{Kuo75}.)\nMore generally, in order for the subsequent theory to make sense, one would\nnaturally need\nto require that $\\mathfrak{g}_{CM}$ be a Lie subalgebra of $\\mathfrak{g}$, that is, for\nthe restriction of the Lie bracket to $\\mathfrak{g}_{CM}$ to preserve\n$\\mathfrak{g}_{CM}$.  As the proofs in the sequel rely strongly on the bracket\nbeing Hilbert-Schmidt, it would be then necessary to add the Hilbert-Schmidt hypothesis as it \ndoes not follow immediately if one only assumes a continuous bracket on\n$\\mathfrak{g}$ which preserves $\\mathfrak{g}_{CM}$.  \n\n\n\n\nAdditionally, the spaces studied in the present paper are well-designed for the study of\ninfinite-dimensional hypoelliptic heat kernel measures, and there has already\nbeen progress on proving quasi-invariance and stronger smoothness properties\nfor these measures in the simplest case of a step two Lie algebra with\nfinite-dimensional center; see \\cite{BGM2013} and\n\\cite{DEM2013}. More generally, the paper \\cite{Pickrell2011} explores related interesting lines of inquiry for heat kernel measures on infinite-dimensional groups, largely in the context of groups of maps from manifolds to Lie groups.\n\n\n\n{\\it Acknowledgements. } The author thanks Bruce Driver and Nathaniel Eldredge for\nhelpful conversations during the writing of this paper.\n\n\n\n\\section{Iterated It\\^o integrals}\n\\label{s.mult}\n\nRecall the standard construction of abstract Wiener spaces.  \nSuppose that $W$ is a real separable Banach space and $\\mathcal{B}_{W}$ is\nthe Borel $\\sigma$-algebra on $W$.\n\n\\begin{defn}\n\\label{d.2.1} \nA measure $\\mu$ on $(W,\\mathcal{B}_{W})$ is called a (mean zero,\nnon-degenerate) {\\em Gaussian measure} provided that its characteristic\nfunctional is given by\n\\[\n\\hat{\\mu}(u) := \\int_W e^{iu(x)} d\\mu(x)\n\t= e^{-\\frac{1}{2}q(u,u)}, \\qquad \\text{ for all } u\\in W^*,\n\\]\nfor $q=q_\\mu:W^*\\times W^*\\rightarrow\\mathbb{R}$ a symmetric, positive\ndefinite quadratic form.\nThat is, $q$ is a real inner product on $W^*$.\n\\end{defn}\n\n\n\n\\begin{thm}\n\\label{t.2.3}\nLet $\\mu$ be a Gaussian measure on $W$.  \nFor $w\\in W$, let\n\\[ \n\\|w\\|_H := \\sup\\limits_{u\\in W^*\\setminus\\{0\\}}\\frac{|u(w)|}{\\sqrt{q(u,u)}},\n\\]\nand define the {\\em Cameron--Martin subspace} $H\\subset W$ by\n\\[ H := \\{h\\in W : \\|h\\|_H < \\infty\\}. \\]\nThen $H$ is a dense subspace of $W$, and there exists a unique inner\nproduct $\\langle\\cdot,\\cdot\\rangle_H$ on $H$ such that $\\|h\\|_H^2\n= \\langle h,h\\rangle_H$ for all $h\\in H$, and $H$ is a separable\nHilbert space with respect to this inner product.  For any $h\\in\nH$, $\\|h\\|_W \\le C \\|h\\|_H$ for some $C<\\infty$.  \n\\end{thm}\n\nAlternatively, given $W$ a\nreal separable Banach space and $H$ a real separable Hilbert space \ncontinuously embedded in $W$ as a dense subspace, then for each $w^*\\in W^*$ there\nexists a unique $h_{w^*}\\in H$ such that $\\langle h,w^*\\rangle = \\langle h,\nh_{w^*}\\rangle_H$ for all $h\\in H$.  Then $W^*\\ni w^*\\mapsto h_{w^*}\\in\nH$ is continuous, linear, and one-to-one with a dense range\n\\begin{equation}\n\\label{e.H*} \nH_* := \\{h_{w^*}:w^*\\in W^*\\},\n\\end{equation}\nand $W^*\\ni w^*\\mapsto h_{w^*}\\in\nW$ is continuous. A Gaussian measure on $W$ is a Borel\nprobability measure $\\mu$ such that, for each $w^*\\in W^*$, the random\nvariable $w\\mapsto\\langle w,w^*\\rangle$ under $\\mu$ is a centered Gaussian\nwith variance $\\|h_{w^*}\\|_H^2$.\n\n\n\n\nSuppose that $P:H\\rightarrow H$ is a finite rank orthogonal projection\nsuch that $PH\\subset H_*$. Let $\\{h_j\\}_{j=1}^m$ be an orthonormal basis for\n$PH$.  Then we may extend $P$ to a (unique) continuous operator\nfrom $W\\rightarrow H$ (still denoted by $P$) by letting\n\\begin{equation}\n\\label{e.proj}\nPw := \\sum_{j=1}^m \\langle w, h_j\\rangle_H  h_j\n\\end{equation}\nfor all $w\\in W$.  \n\\begin{nota}\n\\label{n.proj}\nLet $\\mathrm{Proj}(W)$ denote the collection of finite rank projections\non $W$ such that $PW\\subset H_*$ and $P|_H:H\\rightarrow H$ is an orthogonal\nprojection, that is, $P$ has the form given in equation\n\\eqref{e.proj}.\n\\end{nota}\n\nLet $\\{B_t\\}_{t\\ge0}$ be a Brownian motion on $W$ with variance determined by\n\\[\n\\mathbb{E}\\left[\\langle B_s,h\\rangle\n\t\t_{H} \n\t\t\\langle B_t,k\\rangle\n\t\t_H\\right] \n\t= \\langle h,k \\rangle_{H} \\min(s,t),\n\\]\nfor all $s,t\\ge0$ and $h,k\\in H_*$, where $H_*$ is as in (\\ref{e.H*}).  Note that for any $P\\in\\mathrm{Proj}(W)$, $PB$ is a Brownian motion on $PH$. In the rest of this section, we will verify the existence of martingales defined as certain iterated stochastic integrals with respect to $B_t$.\n\n\nThe following is Proposition 4.1 of \\cite{Melcher2009}.  Note that again this\nwas stated in the context where $H=\\mathfrak{g}_{CM}$ was a ``semi-infinite Lie algebra'', but \na brief inspection of the proof shows that this is a general statement about stochastic\nintegrals on Hilbert spaces.\n\n\\begin{prop}\n\\label{p.int1}\nLet $\\{P_m\\}_{m=1}^\\infty\\subset\\mathrm{Proj}(W)$ such that $P_m|_H\\uparrow\nI_H$.  Then, for $\\xi\\in L^2(\\Delta_n(t),H^{\\otimes n})$ a continuous \nmapping, let\n\\begin{align*}\nJ_n^m(\\xi)_t &:= \\int_{\\Delta_n(t)} \\langle P_m^{\\otimes n} \\xi(s), dB_{s_1}\n\t\t\\otimes\\cdots\\otimes dB_{s_n}\n\t\t\\rangle_{H^{\\otimes n}} \\\\\n\t&= \\int_{\\Delta_n(t)} \\langle \\xi(s), dP_m B_{s_1}\n\t\\otimes\\cdots\\otimes dP_m B_{s_n}\n\t\\rangle_{H^{\\otimes n}}. \n\\end{align*}\nThen $\\{J_n^m(\\xi)_t\\}_{t\\ge0}$ is a continuous $L^2$-martingale, \nand there exists a\ncontinuous $L^2$-martingale $\\{J_n(\\xi)_t\\}_{t\\ge0}$ such that\n\\begin{equation} \n\\label{e.Jnm}\n\\lim_{m\\rightarrow\\infty} \\mathbb{E}\\left[ \\sup_{\\tau\\le t} \n\t|J_n^m(\\xi)_\\tau-J_n(\\xi)_\\tau|^2 \\right] = 0\n\\end{equation}\nand  \n\\begin{equation} \n\\label{e.xi}\n\\mathbb{E}|J_n(\\xi)_t|^2\\le\n\t\\|\\xi\\|^2_{L^2(\\Delta_n(t),H^{\\otimes n})}\n\\end{equation}\nfor all $t<\\infty$.  The process\n$J_n(\\xi)$ is well-defined independent of the choice of increasing\northogonal projections $\\{P_m\\}_{m=1}^\\infty$ into $H_*$, and so will be denoted by\n\\[ J_n(\\xi)_t \n\t= \\int_{\\Delta_n(t)} \\langle \\xi(s), dB_{s_1}\\otimes\\cdots\\otimes dB_{s_n}\n\t\\rangle_{H^{\\otimes n}}.\n\\]\n\\end{prop}\n\n\\iffalse\n\\begin{proof} \nNote first that, \n\\[ J_n^m(\\xi)_t = \\sum_{i_1,\\ldots,i_n=1}^m \n\t\\int_{\\Delta_n(t)} \\langle\\xi(s),h_{i_1}\\otimes\\cdots\\otimes\n\th_{i_n}\\rangle_{H^{\\otimes n}}\n\tdB_{s_1}^{i_1}\\cdots dB_{s_n}^{i_n} \n\\]\nwhere $\\{B^i\\}_{i=1}^m$ are independent real\nvalued Brownian motions.  Let $\\xi_{i_1,\\ldots,i_n} :=\n\\langle\\xi,h_{i_1}\\otimes\\cdots\\otimes h_{i_n}\\rangle$.  Then\n\\[ |\\xi_{i_1,\\ldots,i_n}(s) |^2\\le \\|\\xi(s)\\|^2\n\t_{H^{\\otimes n}} \\]\nand $\\xi_{i_1,\\ldots,i_n} \\in L^2(\\Delta_n(t))$.  Thus, $J_n^m(\\xi)_t$ is \ndefined as a\n(finite dimensional) vector-valued multiple Wiener-It\\^o integral, see for\nexample \\cite{Ito51,Shigekawa04}.\n\nNow note that\n\\begin{align*} \ndJ&_n^m(\\xi)_t \n\t= \\int_{\\Delta_{n-1}(t)} \\langle \\xi(s_1,\\ldots,s_{n-1},t),\n\t\tdP_mB_{s_1}\\otimes\\cdots\\otimes dP_mB_{s_{n-1}} \\otimes dP_mB_t\n\t\t\\rangle_{H^{\\otimes n}} \\\\\n\t&= \\sum_{i=1}^m \\int_{\\Delta_{n-1}(t)} \\langle \\xi(s_1,\\ldots,s_{n-1},t),\n\t\tdP_mB_{s_1}\\otimes\\cdots\\otimes dP_mB_{s_{n-1}} \\otimes h_i\n\t\t\\rangle_{H^{\\otimes n}} dB_t^i.\n\\end{align*}\nThus, the quadratic variation $\\langle J_n^m(\\xi) \\rangle_t$ is given by\n\\[ \\sum_{i=1}^m \\int_0^t \n\t\\bigg|\\int_{\\Delta_{n-1}(\\tau)}\n\t\\langle \\xi(s_1,\\ldots,s_{n-1},\\tau),\n\tdP_mB_{s_1}\\otimes\\cdots\\otimes\n\tdP_mB_{s_{n-1}}\\otimes h_i \\rangle\n\t_{H^{\\otimes n}}\\bigg|^2 d\\tau,\n\\]\nand\n\\begin{align*}\n\\mathbb{E}&|J_n^m(\\xi)_t|^2 \n\t= \\mathbb{E} \\langle J_n^m(\\xi) \\rangle_t \\\\\n\t&= \\sum_{i_1=1}^m \\int_0^t \n\t\t\\mathbb{E}\\bigg[\\sum_{i_2=1}^m \\int_0^{\\tau_1}\n\t\t\\bigg|\\int_{\\Delta_{n-2}(\\tau_2)}\n\t\t\\langle \\xi(s_1,\\ldots,s_{n-2},\\tau_2,\\tau_1),\n\t\tdP_mB_{s_1}\\otimes\\cdots \\\\\n\t&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad \n\t\t\\cdots \\otimes dP_mB_{s_{n-2}}\n\t\t\\otimes h_{i_2}\\otimes h_{i_1}\\rangle\n\t\t_{H^{\\otimes n}}\\bigg|^2d\\tau_2 \\bigg]d\\tau_1.\n\\end{align*}\nIterating this procedure $n$ times gives\n\\begin{align*}\n\\mathbb{E}|J_n^m(\\xi)_t|^2 \n\t&= \\sum_{i_1,\\ldots,i_n=1}^m \\int_{\\Delta_n(t)} \n\t\t\\left|\\langle \\xi(\\tau_1,\\cdots,\\tau_n), \n\t\th_{i_1}\\otimes\\cdots\\otimes h_{i_n} \\rangle\n\t\t_{H^{\\otimes n}}\\right|^2 d\\tau_1\\cdots d\\tau_n \\\\\n\t&\\notag\n\t= \\int_{\\Delta_n(t)} \\|P_m^{\\otimes n} \\xi(s) \\|\n\t\t_{H^{\\otimes n}}^2\n\t\\le \\|\\xi \\|_{L^2(\\Delta_n(t),H^{\\otimes n})}^2,\n\\end{align*}\nand thus, for each $n$, $J_n^m(\\xi)_t$ is bounded uniformly in $L^2$\nindependent of $m$.  \n\n\nNow, for $P\\in\\mathrm{Proj}(W)$, let \n$\nJ_n^P(\\xi)_t\n\t:= \\int_{\\Delta_n(t)} \\langle P^{\\otimes n} \\xi(s), d B_{s_1}\n\t\\otimes\\cdots\\otimes d B_{s_n}\n\t\\rangle_{H^{\\otimes n}}. \n$\nFor $P,Q\\in\\mathrm{Proj}(W)$, a similar argument to the above implies that \n\\begin{equation}\n\\label{e.gub}\n\\mathbb{E}|J_n^P(\\xi)_t - J_n^Q(\\xi)_t|^2 \t\n\t= \\int_{\\Delta_n(t)} \\|P^{\\otimes n} \\xi(s) - Q^{\\otimes n}\n\t\t\\xi(s)\\|^2_{H^{\\otimes n}}\\,ds.\n\\end{equation}\nIn particular, take $P=P_m$ and $Q=P_\\ell$ with $\\ell\\le m$, and note that\n\\begin{align}\n\\notag\n&\\|P_m^{\\otimes n} \\xi(s) - P_\\ell^{\\otimes n}\n\t\t\\xi(s)\\|^2_{H^{\\otimes n}} \\\\\n\t&\\notag\\quad= \\sum_{i_1,\\ldots,i_n=1}^\\infty\n\t\t\\bigg| \\sum_{j=1}^n \\langle P_m^{\\otimes j-1} \\otimes \n\t\t(P_m-P_\\ell)\\otimes P_\\ell^{n-j-1}\\xi(s), h_{i_1}\\otimes\n\t\t\\cdots\\otimes\n\t\th_{i_n} \\rangle_{H^{\\otimes n}}\\bigg|^2 \\\\\n\t&\\label{e.gab}\n\t\\quad\\le n  \\sum_{j=1}^n \\sum_{i_1,\\ldots,i_n=1}^\\infty\n\t\t\\bigg| \\langle P_m^{\\otimes j-1} \\otimes \n\t\t(P_m-P_\\ell)\\otimes P_\\ell^{n-j-1}\\xi(s), h_{i_1}\\otimes\n\t\t\\cdots\\otimes\n\t\th_{i_n} \\rangle_{H^{\\otimes n}}\\bigg|^2 \\\\\n\t&\\notag\n\t\\quad= n \\sum_{j=1}^n \\sum_{i_1,\\ldots,i_{j-1}=1}^m \\sum_{i_j=\\ell+1}^m\n\t\t\\sum_{i_{j+1},\\ldots,i_n=1}^\\ell \n\t\t\\left|\\langle \\xi(s), h_{i_1}\\otimes\\cdots\\otimes h_{i_n} \\rangle\n\t\t_{H^{\\otimes n}}\\right|^2 \n\t\\rightarrow 0,\n\\end{align}\nas $\\ell,m\\rightarrow\\infty$, for all $s\\in\\Delta_n(t)$, since \n$\\|\\xi\\|_{L^2(\\Delta_n(t),H^{\\otimes n})}^2<\\infty$.\nThus, equation (\\ref{e.gub}) implies that\n\\[ \\lim_{\\ell,m\\rightarrow \\infty}\n\t\\mathbb{E}\\left|J_n^m(\\xi)_t - J_n^\\ell(\\xi)_t \\right|^2\n\t= 0, \\]\nby dominated convergence, and $\\left\\{J_n^m(\\xi)_t\\right\\}_{m=1}^\\infty$ \nis Cauchy in $L^2$.  \nSince the space of continuous $L^2$-martingales is complete in the\nnorm $M\\mapsto \\mathbb{E}|M_t|^2$, there exists a continuous martingale\n$\\{J_n(\\xi)_t\\}_{t\\ge0}$ such that\n\\[\n\\lim_{m\\rightarrow\\infty} \\mathbb{E}|J_n^m(\\xi)_t-J_n(\\xi)_t|^2 = 0. \n\\]\nCombining this with Doob's\nmaximal inequality proves equation (\\ref{e.Jnm}).\n\n\nTo see that $J_n(\\xi)_t$ is independent of the choice of basis, suppose now that\n$\\{h'_j\\}_{j=1}^\\infty\\subset H_*$ is another orthonormal\nbasis for $H$, and let $P_m':W\\rightarrow H_*$ \nbe the corresponding orthogonal projections.\nConsider the inequality (\\ref{e.gab}) with $P_\\ell$ replaced by $P_m'$.\nWriting $P_m-P_m'=(P_m-I) + (I-P_m')$, and \nconsidering terms for each fixed $j$, we have \n\\begin{align*}\n& \\sum_{i_1,\\ldots,i_n=1}^\\infty\n\t\t\\bigg| \\langle P_m^{\\otimes j-1} \\otimes \n\t\t(P_m-I)\\otimes(P_m')^{n-j-1}\\xi(s), h_{i_1}\\otimes\n\t\t\\cdots\\otimes\n\t\th_{i_n} \\rangle_{H^{\\otimes n}}\\bigg|^2 \\\\\n\t&= \\sum_{i_1,\\ldots,i_{j-1}=1}^m \\sum_{i_j=m+1}^\\infty\n\t\t\\sum_{i_{j+1},\\ldots,i_n=1}^\\infty\n\t\t\\bigg|\\langle \\xi(s), h_{i_1}\\otimes\n\t\t\\cdots \\otimes h_{i_j} \n\t\t\\otimes P_m' h_{i_{j+1}}\n\t\t\\otimes\\cdots\\otimes P_m' h_{i_n} \\rangle\n\t\t_{H^{\\otimes n}}\\bigg|^2 \\\\\n\t&\\le \\sum_{i_1,\\ldots,i_{j-1}=1}^m \\sum_{i_j=m+1}^\\infty\n\t\t\\sum_{i_{j+1},\\ldots,i_n=1}^\\infty\n\t\t\\left|\\langle \\xi(s), h_{i_1}\\otimes\n\t\t\\cdots\\otimes h_{i_n} \\rangle\n\t\t_{H^{\\otimes n}}\\right|^2\n\t\\rightarrow 0,\n\\end{align*}\nas $m\\rightarrow\\infty$.  Similarly,\n\\begin{align*}\n&\\sum_{i_1,\\ldots,i_n=1}^\\infty\n\t\t\\bigg| \\langle P_m^{\\otimes j-1} \\otimes \n\t\t(I-P_m')\\otimes(P_m')^{n-j-1}\\xi(s), h_{i_1}\\otimes\n\t\t\\cdots\\otimes\n\t\th_{i_n} \\rangle_{H^{\\otimes n}}\\bigg|^2 \\\\\n\t&\\quad= \\sum_{i_1,\\ldots,i_n=1}^\\infty\t\n\t\t\\bigg|\\langle P_m^{\\otimes j-1} \\otimes \n\t\t(I-P_m')\\otimes(P_m')^{n-j-1}\\xi(s),  h_{i_1}'\\otimes\n\t\t\\cdots\\otimes h_{i_n}' \\rangle\n\t\t_{H^{\\otimes n}} \\bigg|^2 \n\t\\rightarrow 0,\n\\end{align*}\nas $m\\rightarrow\\infty$.  Thus,\n\\[ \\lim_{m\\rightarrow\\infty} \\|P_m^{\\otimes n} \\xi(s) - (P_m')^{\\otimes n}\n\t\t\\xi(s)\\|^2_{H^{\\otimes n}} = 0, \\]\nfor each $s\\in\\Delta_n(t)$. \nThus, for $J_n^{m'}(\\xi)_t := J_n^{P_m'}(\\xi)_t$,\nusing equation (\\ref{e.gub}) with $P=P_m$ and $Q=P_m'$ shows that\n\\[ \\lim_{m\\rightarrow\\infty} \\mathbb{E}|J_n^m(\\xi)_t \n\t- J_n^{m'}(\\xi)_t |^2 = 0, \\]\nagain by dominated convergence.  \n\\end{proof}\n\\fi\n\n\n\n\nNow we may use this result to define stochastic integrals taking values in\nanother Hilbert space $K$.\n\n\\begin{prop}\n\\label{p.int2}\nLet $K$ be a Hilbert space and $F\\in L^2(\\Delta_n(t),(H^{\\otimes\nn})^*\\otimes K)$ be a continuous map.  That is, $F:\\Delta_n(t)\\times H^{\\otimes\nn}\\rightarrow K$ is a map continuous in $s$ and linear on $H^{\\otimes n}$ such that\n\\[ \\int_{\\Delta_n(t)} \\|F(s)\\|_2^2\\,ds \n\t= \\int_{\\Delta_n(t)} \\sum_{j_1,\\ldots,j_n=1}^\\infty\n\\|F(s)(h_{j_1}\\otimes\\cdots\\otimes h_{j_n})\\|_K^2\\,ds <\\infty. \\]\nThen\n\\[ J_n^m(F)_t := \\int_{\\Delta_n(t)} F(dP_m B_{s_1}\n\t\\otimes\\cdots\\otimes dP_m B_{s_n}) \n\\]\nis a continuous $K$-valued $L^2$-martingale, and there exists a continuous $K$-valued $L^2$-martingale\n$\\{J_n(F)_t\\}_{t\\ge0}$ such that\n\\begin{equation}\n\\label{e.intF}\n\\lim_{m\\rightarrow\\infty} \\mathbb{E}\\left[ \\sup_{\\tau\\le t} \n\t\\|J_n^m(F)_\\tau-J_n(F)_\\tau\\|_{K}^2 \\right] = 0,\n\\end{equation}\nfor all $t<\\infty$.  The martingale $J_n(F)$ is well-defined independent of \nthe choice of orthogonal projections, \nand thus will be denoted by\n\\[\nJ_n(F)_t = \\int_{\\Delta_n(t)} F(dB_{s_1}\\otimes\\cdots\\otimes dB_{s_n}).\n\\]\n\\end{prop}\n\n\\begin{proof}\nLet $\\{e_j\\}_{j=1}^{\\infty}$ be an orthonormal basis of $K$. \nSince $\\langle F(s)(\\cdot), e_j \\rangle$ is linear on\n$H^{\\otimes n}$, for each $s$ there exists\n$\\xi_j(s)\\in H^{\\otimes n}$ such that\n\\begin{equation}\n\\label{e.xij} \n\\langle \\xi_j(s), k_1\\otimes\\cdots\\otimes k_n \\rangle \n\t= \\langle F(s)(k_1\\otimes\\cdots\\otimes k_n), e_j \\rangle. \n\\end{equation}\nIf $\\xi_j:\\Delta_n(t)\\rightarrow H^{\\otimes n}$ \nis defined by equation (\\ref{e.xij}), \nthen clearly $\\xi_j\\in L^2(\\Delta_n(t),H^{\\otimes n})$ and in particular\n\\begin{align*}\n\\|F\\|_{L^2(\\Delta_n(t)\\times H^{\\otimes n},K)}^2\n\t&=  \\sum_{j=1}^\\infty \\|\\xi_j\\|_{L^2(\\Delta_n(t),H^{\\otimes n})} <\\infty.\n\\end{align*}\nThus, for $J_n(\\xi_j)$ as defined in Proposition \\ref{p.int1},\n\\begin{align*}\n\\mathbb{E}\\left[ \\sum_{j=1}^\\infty |J_n(\\xi_j)_t|^2\\right] \n\t&\\le \\frac{t^n}{n!}\\mathbb{E}\\left[ \\int_{\\Delta_n(t)} \\sum_{j=1}^\\infty |\\langle \\xi_j(s),\n\t\tdB_{s_1}\\otimes\\cdots\\otimes dB_{s_n}\\rangle_{H^{\\otimes n}}|^2\n\t\t\\right] \\\\\n\t&= \\frac{t^n}{n!}\\sum_{j=1}^\\infty \\|\\xi_j\\|^2_{L^2(\\Delta_n(t),H^{\\otimes n})}\n\t<\\infty,\n\\end{align*}\nand so we may write\n\\begin{align*} \n\\sum_{j=1}^\\infty J_n(\\xi_j)_t e_j \n\t&= \\sum_{j=1}^\\infty \\int_{\\Delta_n(t)} \n\t\t\\langle \\xi_j(s),\n\t\tdB_{s_1}\\otimes\\cdots\\otimes dB_{s_n}\\rangle_{H^{\\otimes n}} e_j  \\\\\n\t&= \\int_{\\Delta_n(t)} \\sum_{j=1}^\\infty\n\t\t\\langle F(s)(\n\t\tdB_{s_1}\\otimes\\cdots\\otimes dB_{s_n}),e_j\\rangle_K e_j \\\\\n\t&= \\int_{\\Delta_n(t)} \n\t\tF(s)(dB_{s_1}\\otimes\\cdots\\otimes dB_{s_n}).\n\\end{align*}\nThus, taking $J_n(F)_t := \\sum_{j=1}^\\infty J_n(\\xi_j)_t e_j$,\nwe also have that \n\\begin{align*}\n\\mathbb{E}\\|J_n(F)_t - J_n^m(F)_t\\|_K^2\n\t&= \\mathbb{E}\\left[\\sum_{j=1}^\\infty\n\t\t|J_n(\\xi_j)_t-J_n^m(\\xi_j)_t|^2\\right]\n\t\\rightarrow 0\n\\end{align*}\nas $m\\rightarrow\\infty$ by (\\ref{e.Jnm}) and dominated convergence since\n\\[ \\mathbb{E}|J_n(\\xi_j)_t-J_n^m(\\xi_j)_t|^2 \n\t\\le 4\\|\\xi_j\\|_{L^2(\\Delta_n(t),H^{\\otimes n})}^2 \\]\nby (\\ref{e.xi}).  Then equation (\\ref{e.intF}) holds by Doob's maximal\ninequality.\n\\end{proof}\n\nNote that the preceding results then imply that one may define the above\nstochastic integrals with respect to {\\it any} increasing sequence of\northogonal projections -- that is, we need not require that the projections\nextend continuously to $W$.\n\n\n\\begin{prop}\n\\label{p.bad}\nLet $V$ be an arbitrary finite-dimensional subspace of $H$, and let\n$\\pi:H\\rightarrow V$ denote orthogonal projection onto $V$.  Then for any\nHilbert space $K$ and  $F\\in L^2(\\Delta_n(t),(H^{\\otimes\nn})^*\\otimes K)$ a continuous map, the\nstochastic integral \n\\[ J_n^\\pi(F)_t := \\int_{\\Delta_n(t)} F(d\\pi B_{s_1}\n\t\\otimes\\cdots\\otimes d\\pi B_{s_n}) \n\\]\nis well-defined, and $\\{J_n^\\pi(F)_t\\}_{t\\ge0}$\nis a continuous $K$-valued $L^2$-martingale.  Moreover, if $V_m$ is an increasing sequence of\nfinite-dimensional subspaces of $H$\nsuch that the corresponding orthogonal projections $\\pi_m\\uparrow I_H$, then \n\\begin{equation*} \n\\lim_{m\\rightarrow\\infty} \\mathbb{E}\\left[ \\sup_{\\tau\\le t} \n\t\\|J_n^{\\pi_m}(F)_\\tau-J_n(F)_\\tau\\|^2 \\right] = 0,\n\\end{equation*}\nwhere $J_n(F)$ is as defined in Proposition \\ref{p.int2}.\n\\end{prop}\n\n\\begin{proof}\nFirst consider the case that $K=\\mathbb{R}$, and thus $F(s)=\\langle\n\\xi(s),\\cdot\\rangle$ for a continuous $\\xi\\in L^2(\\Delta_n(t),H^{\\otimes n})$.\nSince $\\pi^{\\otimes n}\\xi\\in L^2(\\Delta_n(t),H^{\\otimes n})$, the definition\nof $J_n^\\pi(\\xi)=J_n(\\pi^{\\otimes n}\\xi)$ follows from Propostion \\ref{p.int1}.  Moreover, by equation\n(\\ref{e.xi}), \n\\begin{align*}\n\\mathbb{E}|J_n^{\\pi_m}(\\xi)_t - J_n(\\xi)_t|^2\n\t&= \\mathbb{E}|J_n(\\pi_m^{\\otimes n}\\xi)_t - J_n(\\xi)_t|^2 \n\t= \\mathbb{E}|J_n(\\pi_m^{\\otimes n}\\xi-\\xi)_t|^2 \\\\\n\t&\\le \\|\\pi_m^{\\otimes n}\\xi-\\xi\\|_{L^2(\\Delta_n(t),H^{\\otimes n})}\n\t\\rightarrow 0\n\\end{align*}\nas $m\\rightarrow\\infty$.  Now the proof for general $F$ follows just as in\nProposition \\ref{p.int2}.\n\\end{proof}\n\n\n\n\n\\section{Abstract nilpotent Lie algebras and groups}\n\\label{s.prelim}\n\n\n\n\\begin{defn}\n\\label{d.semi}\nLet $(\\mathfrak{g},\\mathfrak{g}_{CM},\\mu)$ be an abstract\nWiener space such that $\\mathfrak{g}_{CM}$ is equipped with a nilpotent\nHilbert-Schmidt Lie bracket.  Then we will call\n$(\\mathfrak{g},\\mathfrak{g}_{CM},\\mu)$ an {\\it abstract nilpotent Lie\nalgebra}.\n\\end{defn}\n\n\nThe Baker-Campbell-Hausdorff-Dynkin formula implies that\n\\[ \\log(e^A e^B) = A+B+\\sum_{k=1}^{r-1} \n\t\t\\sum_{(n,m)\\in\\mathcal{I}_k}  \n\t\ta_{n,m}^k\\mathrm{ad}_A^{n_1} \\mathrm{ad}_B^{m_1} \\cdots\n\t\t\\mathrm{ad}_A^{n_k} \\mathrm{ad}_B^{m_k} A,\n\\]\nfor all $A,B\\in\\mathfrak{g}_{CM}$, where \n\\begin{equation*} \na_{n,m}^k := \\frac{(-1)^k}{(k+1)m!n!(|n|+1)}, \n\\end{equation*}\n$\\mathcal{I}_k := \\{(n,m)\\in\\mathbb{Z}_+^k\\times\\mathbb{Z}_+^k : \nn_i+m_i>0 \\text{ for all } 1\\le i\\le k \\}$, and for each multi-index\n$n\\in\\mathbb{Z}_+^k$,\n\\[ n!= n_1!\\cdots n_k! \\quad \\text{ and } \\quad |n|=n_1+\\cdots+n_k; \\]\nsee, for example, \\cite{duiskolk}.  If $\\mathfrak{g}_{CM}$ is nilpotent of step\n$r$, then\n\\[ \\mathrm{ad}_A^{n_1} \\mathrm{ad}_B^{m_1} \\cdots\n\t\t\\mathrm{ad}_A^{n_k} \\mathrm{ad}_B^{m_k} A = 0 \\quad\n\\text{if } |n|+|m|\\ge r. \\]\nfor $A,B\\in\\mathfrak{g}_{CM}$.  \nSince $\\mathfrak{g}_{CM}$ is simply connected and nilpotent, \nthe exponential map is a global diffeomorphism (see, for\nexample, Theorems 3.6.2 of \\cite{Varadarajan} or 1.2.1 of \\cite{CorGrn90}). \nIn particular, we may view $\\mathfrak{g}_{CM}$ as both a Lie algebra and Lie group, and \none may verify that\n\\begin{align}\n\\label{e.mult}\ng\\cdot h &= g+h+\\sum_{k=1}^{r-1} \n\t\t\\sum_{(n,m)\\in\\mathcal{I}_k}  \n\t\ta_{n,m}^k\\mathrm{ad}_g^{n_1} \\mathrm{ad}_h^{m_1} \\cdots\n\t\t\\mathrm{ad}_g^{n_k} \\mathrm{ad}_h^{m_k} g\n\\end{align}\ndefines a group structure on $\\mathfrak{g}_{CM}$.  Note that $g^{-1}=-g$ and \nthe identity $\\mathbf{e}=(0,0)$.\n\n\\begin{defn}\nWhen we wish to emphasize the group structure on $\\mathfrak{g}_{CM}$, we will\ndenote $\\mathfrak{g}_{CM}$ by $G_{CM}$.\n\\end{defn}\n\n\n\n\n\\begin{lem}\n\\label{l.cts}\nThe Banach space topology on $\\mathfrak{g}_{CM}$ makes $G_{CM}$ into a topological group.\n\\end{lem}\n\\begin{proof}\nSince $\\mathfrak{g}_{CM}$ is a topological vector space, \n$g\\mapsto g^{-1}=-g$ and $(g_1,g_2)\\mapsto g_1+g_2$ are continuous by\ndefinition.  The map $(g_1,g_2)\\mapsto [g_1,g_2]$ is continuous in \nthe $\\mathfrak{g}_{CM}$ topology by the boundedness of the Lie bracket.\nIt then follows from (\\ref{e.mult}) that $(g_1,g_2)\\mapsto g_1\\cdot g_2$\nis continuous as well.\n\\end{proof}\n\n\n\n\\label{s.mga}\\subsection{Measurable group actions on $G$}\n\nAs discussed in the introduction, given a Hilbert-Schmidt Lie bracket on\n$\\mathfrak{g}_{CM}$ and a subsequently defined group operation on $G_{CM}$,\none may define a measurable action on $G$ by left or right multiplication by\nan element of $G_{CM}$.  \n\nIn particular, let $\\{e_n\\}_{n=1}^\\infty$ be an orthonormal basis of $\\mathfrak{g}_{CM}$.  \nFor now, fix $n$ and consider the mapping $\\mathfrak{g}_{CM}\\rightarrow \\mathfrak{g}_{CM}^*$ given by $h\\mapsto \\langle\n\\mathrm{ad}_h\\cdot,e_n\\rangle$.  Then this is a continuous linear map on $\\mathfrak{g}_{CM}$\nand in the usual way we may make the identification of\n$\\mathfrak{g}_{CM}^*\\cong \\mathfrak{g}_{CM}$ so that we\ndefine the operator $A_n:\\mathfrak{g}_{CM}\\rightarrow \\mathfrak{g}_{CM}$ given by\n\\[ \\langle A_nh, k\\rangle = \\langle \\mathrm{ad}_h k,e_n\\rangle; \\]\nin particular, $A_nh=\\mathrm{ad}_h^*e_n$.  Note that, for any $h,k\\in \\mathfrak{g}_{CM}$\n\\[ \\langle A_n^*h,k\\rangle  \n\t= \\langle A_nk,h\\rangle \n\t= \\langle \\mathrm{ad}_k^* e_n,h\\rangle\n\t= \\langle e_n , \\mathrm{ad}_k h \\rangle\n\t= - \\langle e_n , \\mathrm{ad}_h k \\rangle\n\t= - \\langle \\mathrm{ad}_h^* e_n,k\\rangle \\]\nand thus $A_n^*=-A_n$.\nNow  fix $h\\in G_{CM}=\\mathfrak{g}_{CM}$.  Then for $\\mathrm{ad}_h:\\mathfrak{g}_{CM}\\rightarrow \\mathfrak{g}_{CM}$ we may write\n\\[\n\\mathrm{ad}_h k = \\sum_n \\langle \\mathrm{ad}_hk, e_n\\rangle e_n\n\t= \\sum_n \\langle A_nh,k\\rangle e_n.\n\\]\nSince each $\\langle A_nh,\\cdot\\rangle\\in \\mathfrak{g}_{CM}^*$ has a measurable linear\nextension to $G$ such that $\\|\\langle\nA_nh,\\cdot\\rangle\\|_{L^2(\\mu)} = \\|A_nh\\|_{\\mathfrak{g}_{CM}}$ \n(see, for example, Theorem 2.10.11 of \\cite{Bogachev1998}), we may extend\n$\\mathrm{ad}_h$ to a measurable linear transformation from $G=\\mathfrak{g}$\nto $G_{CM}=\\mathfrak{g}_{CM}$ (still denoted by $\\mathrm{ad}_h$) given by\n\\[ \n\\mathrm{ad}_h g := \\sum_n \\langle A_nh,g\\rangle e_n. \\]\nNote that here we are using the fact that\n\\begin{align*} \\sum_n \\|\\langle A_nh,\\cdot\\rangle\\|_{L^2(\\mu)}^2\n\t= \\sum_n \\| A_nh\\|_{H}^2\n\t&= \\sum_{n,m} \\langle A_nh,e_m\\rangle^2 \\\\\n\t&= \\sum_{n,m} \\langle \\mathrm{ad}_{h}e_m,e_n\\rangle^2 \n\t\\le \\|h\\|^2 \\|[\\cdot,\\cdot]\\|_{HS}^2\n\\end{align*}\nwhich implies that\n\\[ \\sum_n \\langle A_nh,g\\rangle ^2 < \\infty \\quad g\\text{-a.s.} \\]\nSimilarly, we may define\n\\[ \\mathrm{ad}_g h := -\\mathrm{ad}_h g\n\t= - \\sum_n \\langle A_nh,g\\rangle e_n. \\]\n\nIn a similar way, note that we may write, for $h,k\\in G_{CM}$ and $m< r$,\n\\begin{align*} \n\\mathrm{ad}_h^m k \n\t&= \\sum_{\\ell_1}\\cdots \\sum_{\\ell_m} \\left(\\prod_{b=1}^{m-1} \n\t\t\\langle A_{\\ell_{b}}h, e_{\\ell_{b+1}}\\rangle  \\right)\\langle\n\t\tA_{\\ell_m}h,k\\rangle e_{\\ell_1} \\\\\n\t&= (-1)^m \\sum_{\\ell_1}\\cdots \\sum_{\\ell_m} \\left(\\prod_{b=1}^{m-1} \n\t\t\\langle A_{\\ell_{b}}e_{\\ell_{b+1}},h \\rangle  \\right)\\langle\n\t\tA_{\\ell_m}k,h\\rangle e_{\\ell_1}.\n\\end{align*}\nand thus for $h,k,x\\in G_{CM}$ and $n+m< r$\n\\begin{multline*} \n\\mathrm{ad}_k^n \\mathrm{ad}_h^m x \n\t= (-1)^n \\sum_{j_1}\\cdots \\sum_{j_n}\n\t\t\\sum_{\\ell_1}\\cdots\\sum_{\\ell_m} \n\t\t\\left(\\prod_{a=1}^{n-1} \n\t\t\t\\langle A_{j_{a}}e_{j_{a+1}},k \\rangle  \\right)\n\t\t\\langle A_{j_n}e_{\\ell_1},k\\rangle \\\\\n\t\t\\left(\\prod_{b=1}^{m-1} \n\t\t\t\\langle A_{\\ell_{b}}h, e_{\\ell_{b+1}} \\rangle  \\right)\n\t\t\\langle A_{\\ell_m}h,x\\rangle e_{j_1}.\n\\end{multline*}\n\n\nMore generally for $|n|+|m|<r$\n\\begin{align*} \n&\\mathrm{ad}_k^{n_1} \\mathrm{ad}_h^{m_1}\\cdots \n\t\t\\mathrm{ad}_k^{n_s} \\mathrm{ad}_h^{m_s} k \\\\\n\t&= (-1)^{|n|} \\sum_{j^1_1}\\cdots \\sum_{\\ell^s_{m_s}} \n\t\t\\prod_{c=1}^s \\Bigg\\{\\left(\\prod_{a_c=1}^{n_c-1} \n\t\t\t\\langle A_{j^c_{a_c}}e_{j^c_{a_c+1}},k \\rangle  \\right)\n\t\t\\langle A_{j^c_{n_c}}e_{\\ell^c_1},k\\rangle \\\\\n\t&\\qquad\\qquad\\times\n\t\t\\left(\\prod_{b_c=1}^{m_c-1} \n\t\t\t\\langle A_{\\ell^c_{b_c}}h, e_{\\ell^c_{b_c+1}} \\rangle  \\right)\n\t\t \\Bigg\\}\n\t\t \\left\\{\\prod_{c=1}^{s-1}\\langle A_{\\ell^c_{m_c}}h,e_{j^{c+1}_1}\\rangle\\right\\}\n\t\t\\langle A_{\\ell^s_{m_s}}h,k\\rangle e_{j^1_1}\\end{align*}\nThus, for $h\\in G_{CM}$ and $|n|+|m|<r$, we may define a measurable action on $G$ given by\n\\begin{align*} \nG\\ni g\\mapsto\\,&\\mathrm{ad}_g^{n_1} \\mathrm{ad}_h^{m_1}\\cdots \n\t\t\\mathrm{ad}_g^{n_s} \\mathrm{ad}_h^{m_s} g \\\\\n\t&:= (-1)^{|n|} \\sum_{j^1_1}\\cdots \\sum_{\\ell^s_{m_s}} \n\t\t\\prod_{c=1}^s \\bigg\\{\\left(\\prod_{a_c=1}^{n_c-1} \n\t\t\t\\langle A_{j^c_{a_c}}e_{j^c_{a_c+1}},g \\rangle  \\right)\n\t\t\\langle A_{j^c_{n_c}}e_{\\ell^c_1},g\\rangle \\\\\n\t&\\times\n\t\t\\left(\\prod_{b_c=1}^{m_c-1} \n\t\t\t\\langle A_{\\ell^c_{b_c}}h, e_{\\ell^c_{b_c+1}} \\rangle  \\right)\n\t\t \\Bigg\\}\n\t\t \\left\\{\\prod_{c=1}^{s-1}\\langle A_{\\ell^c_{m_c}}h,e_{j^{c+1}_1}\\rangle\\right\\}\n\t\t\\langle A_{\\ell^s_{m_s}}h,k\\rangle e_{j^1_1}\\in G_{CM}.\n\\end{align*}\n(For $|n|+|m|\\ge r$, we define this mapping to be 0, which is certainly\nmeasurable.)\nAgain, we are using that \n\\begin{multline*} \n\\sum_{j^1_1}\\Bigg\\| \\sum_{j^1_2}\\cdots \\sum_{\\ell^s_{m_s}} \n\t\t\\prod_{c=1}^s \\Bigg\\{\\left(\\prod_{a_c=1}^{n_c-1} \n\t\t\t\\langle A_{j^c_{a_c}}e_{j^c_{a_c+1}},\\cdot \\rangle  \\right)\n\t\t\\langle A_{j^c_{n_c}}e_{\\ell^c_1},\\cdot\\rangle \\\\\n\t\\times\n\t\t\\left(\\prod_{b_c=1}^{m_c-1} \n\t\t\t\\langle A_{\\ell^c_{b_c}}h, e_{\\ell^c_{b_c+1}} \\rangle  \\right)\n\t\t \\Bigg\\}\n\t\t \\left\\{\\prod_{c=1}^{s-1}\\langle A_{\\ell^c_{m_c}}h,e_{j^{c+1}_1}\\rangle\\right\\}\n\t\t\\langle A_{\\ell^s_{m_s}}h,k\\rangle e_{j^1_1} \\Bigg\\|_{L^2(\\mu)}^2 <\\infty.\n\\end{multline*}\nThis holds by straightforward but tedious computations --- in fact, iterative applications of\nCauchy-Schwarz combined with the fact that $\\sum_{n,m} \\|A_ne_m\\|^2 =\n\\|[\\cdot,\\cdot]\\|_{HS}^2<\\infty$. Thus,  \n\\begin{multline*} \n\\sum_{j^1_1}\\Bigg(\\sum_{j^1_2}\\cdots \\sum_{\\ell^s_{m_s}} \n\t\t\\prod_{c=1}^s \\Bigg\\{\\left(\\prod_{a_c=1}^{n_c-1} \n\t\t\t\\langle A_{j^c_{a_c}}e_{j^c_{a_c+1}},g \\rangle  \\right)\n\t\t\\langle A_{j^c_{n_c}}e_{\\ell^c_1},g\\rangle \\\\\n\t\\times\n\t\t\\left(\\prod_{b_c=1}^{m_c-1} \n\t\t\t\\langle A_{\\ell^c_{b_c}}h, e_{\\ell^c_{b_c+1}} \\rangle  \\right)\n\t\t \\Bigg\\}\n\t\t \\left\\{\\prod_{c=1}^{s-1}\\langle A_{\\ell^c_{m_c}}h,e_{j^{c+1}_1}\\rangle\\right\\}\n\t\t\\langle A_{\\ell^s_{m_s}}h,k\\rangle e_{j^1_1}\\Bigg)^2 <\\infty ,\n\\end{multline*}\n$g$-a.s.~and $\\mathrm{ad}_g^{n_1} \\mathrm{ad}_h^{m_1}\\cdots \\mathrm{ad}_g^{n_s}\n\\mathrm{ad}_h^{m_s} g $ as given above is defined a.s.\nThus, we have the following result.\n\n\n\n\\begin{prop}\n\\label{p.gp}\nFor $h\\in G_{CM}$, the mapping $G\\ni g\\mapsto g\\cdot h\\in G$ defined\nanalogously to (\\ref{e.mult}) is a measurable right group action by\n$G_{CM}$ on $G$, and similarly for the left action $g\\mapsto h\\cdot g$.\n\\end{prop}\n\n\\subsection{Examples of abstract nilpotent Lie algebras}\n\n\n\\begin{example}[Free nilpotent Lie algebras]\n\\label{ex.free}\nStarting with an abstract Wiener space $(W,H,\\mu)$, one may construct in the\nstandard way the abstract free\nnilpotent Lie algebra of step $r$ with generators $\\{h_i\\}_{i=1}^\\infty$ an\northonormal basis of $H$.  See for example Section 0 of \\cite{Gromov1996}.\n\\end{example}\n\n\n\\begin{example}[Heisenberg-like algebras]\n\\label{ex.heis}\nLet $(W_i,H_i,\\mu_i)$ for $i=1,2$ be abstract Wiener spaces.  Then for any\n$\\omega:H_1\\times H_1\\rightarrow H_2$ a Hilbert-Schmidt map, we may define a\nLie bracket on $\\mathfrak{g}_{CM}=H_1\\times H_2$ by\n\\[ [(h_1,h_2),(h_1',h_2')] := (0,\\omega(h_1,h_1')), \\]\nand $\\mathfrak{g}=W_1\\times W_2$ may be thought of as an abstract\nHeisenberg-like algebra as in \\cite{DriverGordina2008}.\n\\end{example}\n\nThese abstract Heisenberg-like algebras are central extensions of\none abstract Wiener space by\nanother abstract Wiener space.  The next example generalizes this\nconstruction.\n\n\\begin{example}[Extensions of Lie algebras]\n\\label{ex.ext}\nLet $\\mathfrak{v}$ and $\\mathfrak{h}$ be Lie\nalgebras, and let $\\mathrm{Der}(\\mathfrak{v})$ denote the set of derivations\non $\\mathfrak{v}$; that is, $\\mathrm{Der}(\\mathfrak{v})$ consists of all\nlinear maps $\\rho:\\mathfrak{v}\\rightarrow\\mathfrak{v}$ satisfying Leibniz's\nrule:\n\\[ \\rho([X,Y]_\\mathfrak{v}) = [\\rho(X),Y]_\\mathfrak{v} +\n\t[X,\\rho(Y)]_\\mathfrak{v}. \\]\nNow suppose there are a linear mapping \n$\\alpha: \\mathfrak{h} \\rightarrow \\mathrm{Der}(\\mathfrak{v})$ \nand a skew-symmetric bilinear mapping\n$\\omega:\\mathfrak{h}\\times \\mathfrak{h}\\rightarrow\\mathfrak{v}$, \nsatisfying, for all $X,Y,Z\\in \\mathfrak{h}$,\n\\begin{equation}\n\\label{b1} \n\\tag{B1}\n[\\alpha_X,\\alpha_Y] - \\alpha_{[X,Y]_\\mathfrak{h}} \n\t= \\mathrm{ad}_{\\omega(X,Y)} \n\\end{equation}\nand\n\\begin{equation}\n\\label{b2} \n\\tag{B2}\n\\sum_{\\text{cyclic}}\\left(\\alpha_X \\omega(Y,Z) -\n\t\\omega([X,Y]_\\mathfrak{h},Z)\\right)= 0.  \n\\end{equation}\nThen, one may verify that, for $X_1+V_1,X_2+V_2\\in \\mathfrak{h}\\oplus\n\\mathfrak{v}$,\n\\[ [X_1+V_1, X_2+V_2]_\\mathfrak{g} :=  [X_1,X_2]_\\mathfrak{h} + \\omega(X_1,X_2) \n\t+ \\alpha_{X_1}V_2 -\\alpha_{X_2}V_1 + [V_1,V_2]_\\mathfrak{v} \\]\ndefines a Lie bracket on $\\mathfrak{g}:=\\mathfrak{h}\\oplus\\mathfrak{v}$, \nand we say $\\mathfrak{g}$ is an extension of $\\mathfrak{h}$ over \n$\\mathfrak{v}$. \nThat is, $\\mathfrak{g}$ is the Lie algebra with ideal\n$\\mathfrak{v}$ and quotient algebra\n$\\mathfrak{g}\/\\mathfrak{v}=\\mathfrak{h}$.  The associated exact sequence is\n\\[ 0\\rightarrow\\mathfrak{v}\\overset{\\iota_1}{\\longrightarrow}\\mathfrak{g}\n\t\\overset{\\pi_2}{\\longrightarrow}\\mathfrak{h}\\rightarrow0,\\] \nwhere $\\iota_1$ is\ninclusion and $\\pi_2$ is projection.  In fact,  these are the \nonly extensions of\n$\\mathfrak{h}$ over $\\mathfrak{v}$\n(see, for example, \\cite{AMR00}).\n\n\nNow suppose that $(W,H,\\mu)$ is a real abstract Wiener space, and\n$(\\mathfrak{v},\\mathfrak{v}_{CM},\\mu^0)$ is an abstract nilpotent Lie algebra.\nMotivated by the previous discussion, we may\nconsider $H$ as an abelian Lie algebra and construct extensions of\n$H$ over $\\mathfrak{v}_{CM}$.\nIn this case, we need  a linear mapping\n$\\alpha:H\\rightarrow \\mathrm{Der}(\\mathfrak{v}_{CM})$\nand a skew-symmetric bilinear mapping\n$\\omega:H\\times H\\rightarrow\\mathfrak{v}_{CM}$,\nsuch that $\\omega$ and\n$\\alpha:H\\times\\mathfrak{v}_{CM}\\rightarrow\\mathfrak{v}_{CM}$ \nare both Hilbert-Schmidt and together $\\omega$ and $\\alpha$\nsatisfy (\\ref{b1}) and (\\ref{b2}), which in\nthis setting become\n\\begin{equation*}\n[\\alpha_X,\\alpha_Y] = \\mathrm{ad}_{\\omega(X,Y)}\n\\end{equation*}\nand\n\\begin{equation*}\n\\alpha_X \\omega(Y,Z) + \\alpha_Y \\omega(Z,X) + \\alpha_Z \\omega(X,Y) = 0, \n\\end{equation*}\nfor all $X,Y,Z\\in H$.  Then we may define a Lie algebra structure\non $\\mathfrak{g}_{CM}= H\\oplus \\mathfrak{v}_{CM}$ via the Lie bracket\n\\begin{equation*} \n[(X_1,V_1), (X_2,V_2)]_{\\mathfrak{g}_{CM}} := (0, \\omega(X_1,X_2)\n\t+ \\alpha_{X_1}V_2 - \\alpha_{X_2}V_1 ). \n\\end{equation*}\nAgain, these are in fact the only extensions of $H$\nover $\\mathfrak{v}_{CM}$.  \n\\end{example}\n\nCombining Examples \\ref{ex.heis} and \\ref{ex.ext} shows that these\nconstructions are iterative, in that one may construct new abstract nilpotent\nLie algebras as Lie algebra extensions of another abstract nilpotent Lie\nalgebra.  The next example builds on the previous one to give one precise way\nto construct some Lie algebra extensions.\n\n\n\\begin{example}\n\\label{ex.beta}\nLet $\\beta:H\\rightarrow \\mathfrak{v}_{CM}$ be any Hilbert-Schmidt map, and for\n$X,Y\\in H$ and $V\\in \\mathfrak{v}_{CM}$ define\n\\[ \\omega(X,Y)  := [\\beta(X),\\beta(Y)]_{\\mathfrak{v}_{CM}} \\] \n\\[ \\alpha_X V := \\mathrm{ad}_{\\beta(X)} V = [\\beta(X),V]_{\\mathfrak{v}_{CM}}. \\] \nThen the Lie bracket on $\\mathfrak{g}_{CM}$ is given by\n\\begin{multline*}\n[(X,V),(Y,U)]  \\\\\n\t= (0,[\\beta(X),\\beta(Y)]_{\\mathfrak{v}_{CM}} \n\t+ [\\beta(X),U]_{\\mathfrak{v}_{CM}} - [\\beta(Y),V]_{\\mathfrak{v}_{CM}} \n\t+ [V,U]_{\\mathfrak{v}_{CM}}). \n\\end{multline*}\nNote that, if $\\mathfrak{v}_{CM}$ is nilpotent of step $r$, then\n$\\mathfrak{g}_{CM}$ will automatically be nilpotent of step $r$. \n\\end{example}\n\n\nThe examples above demonstrate that the space of abstract nilpotent Lie\nalgebras is quite rich, and there are many natural examples with a straightforward construction.  This\nsignificantly improves the results of \\cite{Melcher2009}, which studied \nheat kernel measures on nilpotent extensions of abstract\nWiener spaces over {\\it finite-dimensional} nilpotent Lie algebras.\nFor example, the restriction $\\mathrm{dim}(\\mathfrak{v}_{CM})<\\infty$\ntrivializes Example \\ref{ex.beta}.  We elaborate in the following remark.\n\n\n\\begin{remark}\nReturning to Example \\ref{ex.beta}, since $\\beta$ is linear and continuous, we have the decomposition\n\\[ H = \\mathrm{Nul}(\\beta) \\oplus \\mathrm{Nul}(\\beta)^\\perp, \\] \nwhere $\\mathrm{dim}(\\mathrm{Nul}(\\beta)^\\perp) \\le\n\\mathrm{dim}(\\mathfrak{v}_{CM})$.  Thus, for $X,Y\\in H$ we may write\n$X=X_1+X_2, Y=Y_1+Y_2\\in \\mathrm{Nul}(\\beta) \\oplus \\mathrm{Nul}(\\beta)^\\perp$\nand\n\\begin{align*} \n[(X_1+X_2,0),(Y_1+Y_2,0)] \n\t&= [ \\beta(X_1+X_2),\\beta(Y_1+Y_2)] \n\t= [\\beta(X_2),\\beta(Y_2)], \n\\end{align*}\nand $\\omega$ is a map on $\\mathrm{Nul}(\\beta)^\\perp\\times\n\\mathrm{Nul}(\\beta)^\\perp$.  \nThus, $[\\mathrm{Nul}(\\beta),\\mathrm{Nul}(\\beta)]=\\{0\\}$ and similarly\n$[\\mathrm{Nul}(\\beta),\\mathfrak{v}]=\\{0\\}$.  So\n\\[ \\mathfrak{g}_{CM}= H\\oplus\\mathfrak{v}_{CM} = \\mathrm{Nul}(\\beta) \\oplus\n\t\\mathrm{Nul}(\\beta)^\\perp\\oplus\\mathfrak{v}_{CM}, \\]\nand, in particular, when $\\mathrm{dim}(\\mathfrak{v}_{CM})<\\infty$, $\\mathfrak{g}_{CM}$ \nis just an extension of the finite-dimensional vector space\n$\\mathrm{Nul}(\\beta)^\\perp$ by the finite-dimensional Lie algebra\n$\\mathfrak{v}_{CM}$, and the construction is not truly infinite-dimensional.\n\\end{remark}\n\n\n\n\n\\subsection{Properties of $\\mathfrak{g}_{CM}$}\n\n\nThis section collects some results for\ntopological and geometric properties of $\\mathfrak{g}_{CM}$ that we'll require for\nthe sequel.  \n\n\\begin{prop}\n\\label{p.normest}\nFor all $m\\ge2$, $[[[\\cdot,\\cdot],\\ldots],\\cdot]:\\mathfrak{g}_{CM}^{\\otimes m}\n\\rightarrow \\mathfrak{g}_{CM}$ is Hilbert-Schmidt.\n\\end{prop}\n\n\\begin{proof}\nFor $m=2$, this follows from the definition of $G$.  Now assume the statement holds for all $m=\\ell$, and \nconsider $m=\\ell+1$.  Writing $[[h_{i_1},h_{i_2}],\\cdots,h_{i_\\ell+1}]\\in \n\\mathfrak{g}_{CM}$ in terms of the orthonormal basis\n$\\{e_j\\}_{j=1}^\\infty$ and using \nH\\\"older's inequality gives\n\\begin{align*}\n\\|[[[\\cdot,&\\cdot],\\ldots],\\cdot]\\|_2^2\n\t= \\|[[[\\cdot,\\cdot],\\ldots],\\cdot]\\|_{(\\mathfrak{g}_{CM}^*)^{\\otimes \\ell+1}\n\t\t\\otimes\\mathfrak{g}_{CM}} \\\\\n\t&= \\sum_{i_1,\\ldots,i_{\\ell+1}=1}^\\infty \n\t\t\\|[[[h_{i_1},h_{i_2}],\\cdots,h_{i_\\ell}],h_{i_{\\ell+1}}]\\|\n\t\t^2 \\\\\n\t&= \\sum_{i_1,\\ldots,i_{\\ell+1}=1}^\\infty \\left\\|\\sum_{j=1}^\\infty \n\t\t[e_j,h_{i_{\\ell+1}}]\n\t\t\\langle e_j, [[h_{i_1},h_{i_2}],\\cdots,h_{i_\\ell}]\\rangle\n\t\t\\right\\|^2 \\\\\n\t&\\le \\sum_{i_1,\\dots,i_{\\ell+1}=1}^\\infty \\left(\\sum_{j=1}^\\infty \n\t\t\t\\| [e_j,h_{i_{\\ell+1}}] \\|^2\\right)\n\t\t\\left(\\sum_{j=1}^\\infty \n\t\t|\\langle e_j,[[h_{i_1},h_{i_2}],\\cdots,h_{i_\\ell}]\\rangle|^2\\right) \\\\\n\t&= \\left(\\sum_{i_{\\ell+1}=1}^\\infty \\sum_{j=1}^\\infty \n\t\t\t\\| [e_j,h_{i_{\\ell+1}}] \\|^2\\right)\n\t\t\\left(\\sum_{i_1,\\dots,i_\\ell=1}^\\infty \n\t\t\\|[[h_{i_1},h_{i_2}],\\cdots,h_{i_\\ell}]\\|^2\\right)\n\t\t\\\\\n\t&\\le \\|[\\cdot,\\cdot]\\|_{(\\mathfrak{g}_{CM}^*)^{\\otimes 2}\n\t\t\\otimes\\mathfrak{g}_{CM}}^2 \\|[[[\\cdot,\\cdot],\\ldots],\\cdot]\\|_{(\\mathfrak{g}_{CM}^*)^{\\otimes \\ell}\n\t\t\\otimes\\mathfrak{g}_{CM}}^2\n\\end{align*}\nand the last line is finite by the induction hypothesis.\n\\end{proof}\n\nNext we will recall that the flat and geometric topologies on\n$\\mathfrak{g}_{CM}$ are equivalent.  First we set the following notation.\n\n\\begin{notation}\nFor $g\\in G_{CM},$ let $L_g:G_{CM}\\rightarrow G_{CM}$ and $R_g:G_{CM}\\rightarrow G_{CM}$\ndenote left and right multiplication by $g$, respectively.  As $G_{CM}$\nis a vector space, to each $g\\in G_{CM}$ we can associate the tangent\n\tspace $T_g G_{CM}$ to $G_{CM}$ at $g$, which is naturally isomorphic to $G_{CM}$.\nFor $f:G_{CM}\\rightarrow\\mathbb{R}$ a Frech\\'{e}t smooth function and  \n\t$v,x\\in G_{CM}$ and $h\\in\\mathfrak{g}_{CM}$, let\n\\[ f'(x)h := \\partial_h f(x) = \\frac{d}{dt}\\bigg|_{0}f(x+th), \\]\n\tand let $v_x \\in T_x G_{CM}$ denote the tangent vector\nsatisfying $v_xf=f'(x)v$.  If $\\sigma(t)$ is any smooth curve in\n$G_{CM}$ such that $\\sigma(0) = x$ and $\\dot{\\sigma}(0)=v$ (for example,\n$\\sigma(t) = x+tv$), then\n\\[ L_{g*} v_x = \\frac{d}{dt}\\bigg|_0 g\\cdot \\sigma(t). \\]\n\\end{notation}\n\n\\begin{notation}\n\\label{n.length}\nLet $C^1([0,1],G_{CM})$ denote the collection of $C^1$-paths\n$g:[0,1]\\rightarrow G_{CM}$.  The length of $g$ is defined as \n\\[ \\ell_{CM}(g) := \\int_0^T \\|L_{g^{-1}(s)*}g'(s)\\|_{\\mathfrak{g}_{CM}}\\,ds.\n\\]\nThe Riemannian distance between $x,y\\in G_{CM}$ is then defined as\n\\begin{align*} d(x,y) := \\inf\\{\\ell_{CM}(g): g\\in C_h^1([0,1],G_{CM}) \\text{ such that }\n\tg(0)=x \\text{ and } g(1)=y \\}. \n\\end{align*}\n\\end{notation}\n\nThe following proposition was proved as Corollary 4.13 in\n\\cite{Melcher2009}.  This was proved under the conditions that\n$\\mathfrak{g}_{CM}$ was a ``semi-infinite Lie algebra'', that is,\nunder the assumption that $\\mathfrak{g}_{CM}$ was a nilpotent Lie algebra\nextension (as in Example \\ref{ex.ext}) \nof a finite-dimensional nilpotent Lie algebra $\\mathfrak{v}$ by an\nabstract Wiener space.  However, a cursory inspection of the proofs\nthere will show that they only depended on the fact that the Lie\nbracket was Hilbert-Schmidt, and not on the ``stratified'' structure\nof $\\mathfrak{g}_{CM}=H\\times\\mathfrak{v}$ or the fact that the image of the Lie bracket\nwas a finite-dimensional subspace.\n\n\n\n\\begin{prop}\n\\label{p.length}\nThe topology on $G_{CM}$ induced by $d$ is equivalent to the Hilbert\ntopology induced by $\\|\\cdot\\|_{\\mathfrak{g}_{CM}}$.\n\\end{prop}\n\n\n\nWe may find a uniform lower bound on the Ricci curvature of all\nfinite-dimensional subgroups of $G_{CM}$.  Finite-dimensional subgroups\nof $G_{CM}$ may be obtained by taking the Lie algebra\ngenerated by any finite-dimensional subspace of $\\mathfrak{g}_{CM}$\n(which is again necessarily finite-dimensional by the nilpotence\nof the bracket) and endowing it with the standard group operation via the\nBaker-Campbell-Hausdorff-Dynkin formula.   An analogue of the following proposition was\nproved as Proposition 3.23 and Corollary 3.24 in \\cite{Melcher2009}.\nIt follows directly from the form of the Ricci curvature on nilpotent\ngroups (endowed with a left invariant metric) and the assumption\nthat the Lie bracket is Hilbert-Schmidt.  This proof is essentially the same as in\n\\cite{Melcher2009}, but it is quite brief and so is included for completeness.\n\n\\begin{prop}\n\\label{p.Ric}\nLet \n\\[ k := -\\frac{1}{2} \\sup \\left\\{ \n\t\t\\|[\\cdot,X]\\|^2_{\\mathfrak{g}_{CM}^*\\otimes\\mathfrak{g}_{CM}}   \n\t\t: \\,\\|X\\|_{\\mathfrak{g}_{CM}}=1 \\right\\}. \\]\nThen $k>-\\infty$ and $k$ is the largest constant such that\n\\[ \\langle \\mathrm{Ric}^\\pi X,X\\rangle_{\\mathfrak{g}_\\pi} \\ge \n\tk \\|X\\|^2_{\\mathfrak{g}_\\pi}, \n\t\\quad \\text{ for all } X \\in\\mathfrak{g}_\\pi, \\]\nholds uniformly for all $\\mathfrak{g}_\\pi$ finite-dimensional Lie subalgebras\nof $\\mathfrak{g}_{CM}$.\n\\end{prop}\n\n\\begin{proof}\nFor $\\mathfrak{g}$ any nilpotent Lie algebra with orthonormal basis\n$\\Gamma$, \n\\begin{align*}\n\\langle \\mathrm{Ric}\\, X,X\\rangle \n\t&= \\frac{1}{4}\\sum_{Y\\in\\Gamma} \\|\\mathrm{ad}^*_Y X\\|^2 \n\t\t- \\frac{1}{2}\\sum_{Y\\in\\Gamma} \\|\\mathrm{ad}_Y X\\|^2 \n\t\\ge - \\frac{1}{2}\\sum_{Y\\in\\Gamma} \\|[Y,X]\\|^2\n\\end{align*}\nfor all $X\\in\\mathfrak{g}$.  Thus, for $\\mathfrak{g}_\\pi$ any\nfinite-dimensional Lie algebra\n\\[ \\langle \\mathrm{Ric}^\\pi X,X\\rangle_{\\mathfrak{g}_\\pi} \\ge \n\tk_\\pi \\|X\\|^2_{\\mathfrak{g}_\\pi}, \n\t\\quad \\text{ for all } X \\in\\mathfrak{g}_\\pi, \\]\nwhere\n\\begin{equation}\n\\label{e.pah}\n k_\\pi :=  - \\frac{1}{2}\\sup \\left\\{ \n\t\t\\|[\\cdot,X]\\|^2_{\\mathfrak{g}_\\pi^*\\otimes\\mathfrak{g}_\\pi}\n \t\t:\\, \\|X\\|_{\\mathfrak{g}_\\pi}=1 \\right\\} \n\t\\ge - \\frac{1}{2}\\|[\\cdot,\\cdot]\\|_2^2 > -\\infty.\n\\end{equation}\nTaking the infimum of $k_\\pi$ over all $\\mathfrak{g}_\\pi$ completes the proof.\n\\end{proof}\n\n\n\\section{Brownian motion on $G$}\n\\label{s.BM}\nSuppose that $B_t$ is a smooth curve in $\\mathfrak{g}_{CM}$ with\n$B_0=0$, and consider the differential equation\n\\[ \\dot{g}_t = g_t \\dot{B}_t \n\t:= L_{g_t*}\\dot{B}_t, \\quad \\text{ with } g_0=\\mathbf{e}. \\]  \nThe solution $g_t$ may be written as follows (see\n\\cite{Strichartz87}):  For $t>0$, let $\\Delta_n(t)$ denote the simplex in\n$\\mathbb{R}^n$ given by\n\\[ \\{s=(s_1,\\cdots,s_n)\\in\\mathbb{R}^n: 0<s_1<s_2<\\cdots<s_n<t\\}. \\]\nLet $\\mathcal{S}_n$ denote the permutation group on $(1,\\cdots,n)$, and, for each\n$\\sigma\\in \\mathcal{S}_n$, let $e(\\sigma)$ denote the number of ``errors'' in the\nordering $(\\sigma(1),\\sigma(2),\\cdots,\\sigma(n))$, that is, $e(\\sigma)=\\#\n\\{j<n: \\sigma(j)>\\sigma(j+1)\\}$.  Then\n\\begin{multline}\n\\label{e.ode}\ng_t = \\sum_{n=1}^r \\sum_{\\sigma\\in \\mathcal{S}_n} \n\t\\left( (-1)^{e(\\sigma)}\\bigg\/ n^2 \n\t\\begin{bmatrix} n-1 \\\\ e(\\sigma) \\end{bmatrix}\\right) \\times \\\\\n\t\\int_{\\Delta_n(t)} [ \\cdots[\\dot{B}_{s_{\\sigma(1)}},\n\t\\dot{B}_{s_{\\sigma(2)}}],\\ldots,] \\dot{B}_{s_{\\sigma(n)}}]\\, ds,\n\\end{multline}\nwhere the $n=1$ term is understood to be $\\int_0^t dB_s = B_t$.\nUsing this as our motivation, we first explore stochastic integral\nanalogues of equation (\\ref{e.ode}) where the smooth curve $B$ is replaced by\nBrownian motion on $\\mathfrak{g}$.\n\n\n\n\n\n\n\n\\subsection{Brownian motion and finite-dimensional approximations}\n\n\nWe now return to the setting of an abstract Wiener space\n$(\\mathfrak{g},\\mathfrak{g}_{CM},\\mu)$ endowed with a nilpotent\nHilbert-Schmidt Lie bracket on $\\mathfrak{g}_{CM}$.\nAgain, let $B_t$ denote Brownian motion on $\\mathfrak{g}$.\nBy equation (\\ref{e.ode}), the solution to the Stratonovich\nstochastic differential equation\n\\[ \\delta g_t = L_{g_t*} \\delta B_t, \\quad \\text{ with } g_0=\\mathbf{e}, \\]\nshould be given by\n\\begin{equation}\n\\label{e.gt}\ng_t = \\sum_{n=1}^{r} \\sum_{\\sigma\\in\\mathcal{S}_n} c^\\sigma_n \\int_{\\Delta_n(t)} \n\t[ [\\cdots[\\delta B_{s_{\\sigma(1)}},\\delta B_{s_{\\sigma(2)}}],\\cdots], \n\t\\delta B_{s_{\\sigma(n)}}],\n\\end{equation}\nfor coefficients $c_n^\\sigma$ determined by equation (\\ref{e.ode}).\n\nTo understand the integrals in (\\ref{e.gt}), consider the following heuristic\ncomputation.\nLet $\\{M_n(t)\\}_{t\\ge0}$ denote the process in $\\mathfrak{g}^{\\otimes n}$ \ndefined by\n\\[ \nM_n(t) := \\int_{\\Delta_n(t)} \\delta B_{s_1}\\otimes\\cdots\\otimes \\delta\n\tB_{s_n}.\n\\]\nBy repeatedly applying the definition of the Stratonovich integral, \nthe iterated Stratonovich integral $M_n(t)$ \nmay be realized as a linear combination of iterated It\\^o integrals:\n\\[ M_n(t) = \\sum_{m=\\lceil n\/2\\rceil}^n \\frac{1}{2^{n-m}}\n\t\t\\sum_{\\alpha\\in\\mathcal{J}_n^m} I^n_t(\\alpha), \\]\nwhere\n\\[ \\mathcal{J}_n^m := \\left\\{(\\alpha_1,\\ldots,\\alpha_m)\\in\\{1,2\\}^m :\n\t\\sum_{i=1}^m \\alpha_i = n \\right\\}, \\]\nand, for $\\alpha\\in\\mathcal{J}_n^m$, $I_t^n(\\alpha)$ is the iterated \nIt\\^o integral\n\\[ I_t^n(\\alpha) = \\int_{\\Delta_m(t)} dX^1_{s_1}\\otimes\\cdots\\otimes \n\tdX^m_{s_m} \\]\nwith\n\\[ dX_s^i = \\left\\{ \\begin{array}{cl} dB_s & \\text{if } \\alpha_i=1 \\\\\n\t\\sum_{j=1}^\\infty h_j \\otimes h_j \\, ds & \\text{if } \\alpha_i=2\n\t\\end{array} \\right. ; \\]\ncompare with Proposition 1 of \\cite{BenArous89}.  This change from multiple\nStratonovich integrals to multiple It\\^o integrals may also be recognized as a\nspecific case of the Hu-Meyer formulas \\cite{HuMeyer88-2,HuMeyer88-1},\nbut we will compute more explicitly to verify that our integrals are well-defined.\n\n\nDefine $F_1:\\mathfrak{g}_{CM}\\to\\mathfrak{g}_{CM}$ by $F_1(k)=k$, and for $n\\in\\{2,\\cdots,r\\}$ define $F_n:\\mathfrak{g}_{CM}^{\\otimes n}\n\\rightarrow\\mathfrak{g}_{CM}$ by\n\\begin{equation}\n\\label{e.1Fn}\n F_n(k_1\\otimes\\cdots\\otimes k_n) \n\t:= [ [ [\\cdots[k_1,k_2],k_3],\\cdots],k_n]. \n\\end{equation}\nFor each fixed $n$ and $\\sigma\\in\\mathcal{S}_n$, define $F_n^\\sigma:\\mathfrak{g}_{CM}^{\\otimes n}\n\\rightarrow\\mathfrak{g}_{CM}$ by\n\\begin{equation}\n\\label{e.Fn}\n\\begin{split}\nF_n^\\sigma(k_1\\otimes\\cdots\\otimes k_n) \n\t&:= F_n(k_{\\sigma(1)}\\otimes\\cdots\\otimes k_{\\sigma(n)}) \\\\\n\t&= [ [\\cdots[k_{\\sigma(1)},k_{\\sigma(2)}],\\cdots], \n\t\tk_{\\sigma(n)}].\n\\end{split}\n\\end{equation}\nThen we may write\n\\[\ng_t = \\sum_{n=1}^{r} \\sum_{\\sigma\\in\\mathcal{S}_n} \n\tc^\\sigma_n F^\\sigma_n (M_n(t))\n\t= \\sum_{n=1}^{r} \\sum_{\\sigma\\in\\mathcal{S}_n}\n\t\\sum_{m=\\lceil n\/2\\rceil}^n \\frac{c^\\sigma_n }{2^{n-m}}\n\t\t\\sum_{\\alpha\\in\\mathcal{J}_n^m} F^\\sigma_n (I^n_t(\\alpha)),\n\\]\npresuming we can make sense of the integrals $F_n^\\sigma(I_t^n(\\alpha))$.\n\nFor each $\\alpha$, let $p_\\alpha=\\#\\{i:\\alpha_i=1\\}$ and $q_\\alpha=\\#\\{i:\n\\alpha_i=2\\}$ (so that $p_\\alpha+q_\\alpha=m$ when $\\alpha\\in\\mathcal{J}_n^m$),\nand let \n\\[ \\mathcal{J}_n := \\bigcup_{m=\\lceil n\/2\\rceil}^n \\mathcal{J}_n^m. \\]\nThen, for each $\\sigma\\in\\mathcal{S}_n$ and $\\alpha\\in\\mathcal{J}_n$, \n\\[ F_n^\\sigma(I_t^n(\\alpha))\n\t= \\int_{\\Delta_{p_\\alpha}(t)} f_\\alpha(s,t) \\hat{F}_n^{\\sigma,\\alpha}\n\t\t(dB_{s_1}\\otimes\\cdots\\otimes dB_{s_{p_\\alpha}}),\n\\]\nwhere $\\hat{F}_n^{\\sigma,\\alpha}$ and $f_\\alpha$ are as follows.\n\nThe map $\\hat{F}_n^{\\sigma,\\alpha}:\\mathfrak{g}^{\\otimes p_\\alpha}\n\\rightarrow\\mathfrak{g}$ is defined by\n\\begin{multline}\n\\label{e.Fhat}\n\\hat{F}_n^{\\sigma,\\alpha}(k_1\\otimes\\cdots\\otimes k_{p_\\alpha}) \\\\\n\t:= \\sum_{j_1,\\ldots,j_{q_\\alpha}=1}^\\infty \n\t\tF_n^{\\sigma'}(k_1\\otimes\\cdots\\otimes k_{p_\\alpha}\n\t\t\\otimes h_{j_1}\\otimes h_{j_1}\n\t\t\\otimes\\cdots\\otimes h_{j_{q_\\alpha}}\\otimes h_{j_{q_\\alpha}}), \n\\end{multline}\nfor $\\{h_j\\}_{j=1}^\\infty$ an orthonormal basis of $\\mathfrak{g}_{CM}$ and\n$\\sigma'=\\sigma'(\\alpha)\\in\\mathcal{S}_n$ given by \n$\\sigma'=\\sigma\\circ\\tau^{-1}$, for any $\\tau\\in\\mathcal{S}_n$ such that\n\\begin{multline*} \n\\tau(dX^1_{s_1}\\otimes\\cdots\\otimes dX^m_{s_m}) \\\\\n\t= \\sum_{j_1,\\cdots,j_{q_\\alpha}=1}^\\infty dB_{s_1}\\otimes\\cdots\n\t\\otimes dB_{s_{p_\\alpha}}\\otimes h_{j_1}\\otimes h_{j_1}\\otimes\\cdots\n\t\\otimes h_{j_{q_\\alpha}}\\otimes h_{j_{q_\\alpha}} ds_1\\cdots\n\tds_{q_\\alpha}.\n\\end{multline*}\n\nTo define $f_\\alpha$, first consider the polynomial of order $q_\\alpha$,\nin the variables $s_i$ with $i$ such that $\\alpha_i=1$ and in the variable $t$, given by evaluating the integral\n\\begin{equation}\n\\label{e.fprime} \nf_\\alpha'( (s_i:\\alpha_i=1),t)\n\t= \\int_{\\Delta'_{q_\\alpha}(t)} \\prod_{i: \\alpha_i=2} ds_i,\n\\end{equation}\nwhere $\\Delta'_{q_\\alpha}(t)=\\{s_{i-1}<s_i<s_{i+1}: \\alpha_i=2\\}$ with \n$s_0=0$ and $s_{m+1}=t$.  \nThen $f_\\alpha$ is $f_\\alpha'$ with the variables reindexed by the\nbijection $\\{i: \\alpha_i=1\\}\\rightarrow\\{1,\\ldots,p_\\alpha\\}$ that\nmaintains the natural ordering of these sets.\n(For example, for $\\alpha=(1,2,1,2)\\in\\mathcal{J}_6^4$,\n\\[ f_\\alpha'(s_1,s_3,t)=\\int_{\\{s_1<s_2<s_3,s_3<s_4<t\\}} ds_2\\,ds_4\n\t= (t-s_3)(s_3-s_1) , \\]\nso that $f_\\alpha(s_1,s_2,t)=(t-s_2)(s_2-s_1)$.) \n\nThis explicit realization of $f_\\alpha$ is not critical to the sequel.\nIt is really only necessary to know that \n$f_\\alpha$ is a polynomial of order $q_\\alpha$ in\n$s=(s_1,\\ldots,s_{p_\\alpha})$ and $t$, and thus may be written as\n\\[ f_\\alpha(s,t) = \\sum_{a=0}^{q_\\alpha} b_\\alpha^a t^a \n\t\\tilde{f}_{\\alpha,a}(s), \n\\]\nfor some coefficients $b_\\alpha^a\\in\\mathbb{R}$ and polynomials \n$\\tilde{f}_{\\alpha,a}$ of degree $q_\\alpha-a$ in $s$.  \nNow, if $\\hat{F}_n^{\\sigma,\\alpha}$ is\nHilbert-Schmidt on $\\mathfrak{g}_{CM}^{\\otimes p_\\alpha}$, then\n\\[ \\int_{\\Delta_{p_\\alpha}(t)} \\left\\|\\tilde{f}_{\\alpha,a}(s)\n\t\\hat{F}_n^{\\sigma,\\alpha} \\right\\|_2^2\\, ds\t\n\t= \\left\\|\\tilde{f}_{\\alpha,a}\\right\\|^2_{L^2(\\Delta_{p_\\alpha}(t))}\n\t\t\\left\\|\\hat{F}_n^{\\sigma,\\alpha} \\right\\|_2^2\n\t<\\infty,\n\\]\nand \n\\begin{align}\n\\label{e.a}\nF_n^\\sigma(I_t^n(\\alpha))\n\t&= \\sum_{a=0}^{q_\\alpha} b_\\alpha^a t^a J_n(\\tilde{f}_{\\alpha,a}\n\t\t\\hat{F}_n^{\\sigma,\\alpha})_t\n\\end{align}\nmay be understood in the sense of the limit integrals in \nProposition \\ref{p.int2}.  (In particular, if $\\alpha_m=1$, then \n$f_\\alpha=f_\\alpha(s)$ does not depend on $t$, and Proposition \\ref{p.int2} \nimplies that $F_n^\\sigma(I_t^n(\\alpha))$ is a $\\mathfrak{v}$-valued \n$L^2$-martingale.)\n\nThe above computations show that, if for all $n$, $\\sigma\\in\\mathcal{S}_n$,\nand $\\alpha\\in\\mathcal{J}_n$,\n$\\hat{F}_n^{\\sigma,\\alpha}$ is Hilbert-Schmidt, then we may rewrite (\\ref{e.gt}) as\n\\[ g_t = \\sum_{n=1}^{r} \\sum_{\\sigma\\in\\mathcal{S}_n}\n\t\\sum_{m=\\lceil n\/2\\rceil}^n \\frac{c^\\sigma_n }{2^{n-m}}\n\t\t\\sum_{\\alpha\\in\\mathcal{J}_n^m}  \n\t\\sum_{a=0}^{q_\\alpha} b_\\alpha^a t^a J_n(\\tilde{f}_{\\alpha,a}\n\t\t\\hat{F}_n^{\\sigma,\\alpha})_t,\n\\]\nwhere $J_n$ is as defined in Proposition \\ref{p.int2}.  \nThe next proposition shows that  $\\hat{F}_n^{\\sigma,\\alpha}$ is \nHilbert-Schmidt as desired, and thus $g_t$ in (\\ref{e.gt}) is well-defined. \n\n\n\\begin{prop}\n\\label{p.HS}\nLet $n\\in\\{2,\\ldots,r\\}$, $\\sigma\\in\\mathcal{S}_n$, and \n$\\alpha\\in\\mathcal{J}_n$.\nThen $\\hat{F}_n^{\\sigma,\\alpha}:\\mathfrak{g}_{CM}^{\\otimes p_\\alpha}\n\\rightarrow \\mathfrak{g}_{CM}$ is Hilbert-Schmidt.\n\\end{prop}\n\n\\begin{proof}\nFor the whole of this proof, all sums will be taken over an orthonormal basis\nof $\\mathfrak{g}_{CM}$.\n\nNow, for $F_n$ and $F_n^\\sigma$ as defined in equations (\\ref{e.1Fn}) and\n(\\ref{e.Fn}), we may write\n\\begin{align*} \nF_n^\\sigma(k_1\\otimes\\cdots\\otimes k_{p_\\alpha}\\otimes h_1\\otimes\n\t\th_1\\otimes\\cdots\\otimes h_{q_\\alpha}\\otimes h_{q_\\alpha}) \n\t= F_n(A_{\\sigma(1)}\\otimes\\ldots\\otimes A_{\\sigma(n)})\n\\end{align*}\nwhere\n\\[ \nA_b := \\left\\{\\begin{array}{ll}\n\t\tk_b & \\text{if } b=1,\\ldots,p_\\alpha \\\\\n\t\th_{\\lceil (b-p_\\alpha)\/2\\rceil} & \\text{if } b=p_\\alpha + 1,\\ldots,n\n\t\\end{array} \\right. .\n\\] \n\\iffalse\n\\[ \nA_b := \\left\\{\\begin{array}{ll}\n\t\tk_b & \\text{if } b=1,\\ldots,p_\\alpha \\\\\n\t\th_{\\lceil \\ell\/2\\rceil} & \\text{if } b=p_\\alpha + \\ell\n\t\t\t\\text{ for } \\ell=1,\\ldots,2q_\\alpha\n\t\\end{array} \\right. .\n\\]\n\\fi\n\nIf $q_\\alpha=0$, then $p_\\alpha=n$, each $A_{\\sigma(j)}=k_i$ for some\n$i=1,\\ldots,n$, and\n\\begin{align*}\n\\|\\hat{F}_n^{\\sigma,\\alpha}\\|_2^2\n\t&= \\sum_{k_1,\\ldots,k_n} \n\t\t\\|F_n^\\sigma(k_1\\otimes\\cdots\\otimes k_n)\\|_{\\mathfrak{g}_{CM}}^2 \\\\\n\t&= \\sum_{k_1,\\ldots,k_n} \n\t\t\\|F_n(k_1\\otimes\\cdots\\otimes k_n)\\|_{\\mathfrak{g}_{CM}}^2\n\t= \\|[ [[\\ldots[\\cdot,\\cdot],\\ldots],\\cdot]\\|_{(\\mathfrak{g}_{CM}^*)^{\\otimes\n\tn}\\otimes\\mathfrak{g}_{CM}}^2\n\\end{align*}\nwhich is finite by Proposition \\ref{p.normest}.\n\n\nNow, if $q_\\alpha=1$, let $N=N(\\sigma)$ denote the second\n$j$ such that $\\sigma(j)\\in\\{n-1,n\\}$; that is,\n\\begin{align*}\nF_n^\\sigma(k_1\\otimes\\cdots\\otimes& k_{n-1}\\otimes h_1\\otimes\n\t\th_1) \\\\\n\t&= F_n(A_{\\sigma(1)}\\otimes \\cdots\\otimes A_{\\sigma(N-1)}\\otimes h_1 \\otimes\n\t\tA_{\\sigma(N+1)}\\otimes\\ldots\\otimes A_{\\sigma(n)}) \\\\\n\t&= [[[[ [ \\ldots[A_{\\sigma(1)},A_{\\sigma(2)}],\\ldots],A_{\\sigma(N-1)}]\n\t\t,h_1],A_{\\sigma(N+1)}],\\ldots],A_{\\sigma(n)}]\n\\end{align*} \nwhere exactly one of $A_{\\sigma(1)},\\ldots,A_{\\sigma(N-1)}$ is $h_1$ and,\nfor all $j>N$, $A_{\\sigma(j)}=k_i$ for some \n\\[ i\\in I := I(\\sigma):= \\{ \\sigma(j):j=N+1,\\ldots,n\\} \\subseteq\n \\{1,\\ldots,p_\\alpha\\}= \\{1,\\ldots,n-2\\}. \\]\nThus, writing\n\\[ \\mathcal{A}(h_1, k_i : i\\in I^c) \n\t:= [ [ \\ldots[A_{\\sigma(1)},A_{\\sigma(2)}],\\ldots],A_{\\sigma(N-1)}], \\]\nwe have that\n\\begin{multline*}\nF_n^\\sigma(k_1\\otimes\\cdots\\otimes k_{n-1}\\otimes h_1\\otimes h_1) \\\\\n\t= \\sum_{e_1}\n\t\t\\,\\langle \\mathcal{A}(h_1,k_i:i\\in I^c),e_1\\rangle_{\\mathfrak{g}_{CM}}\n\t\t[ [ \\ldots[e_1,h_1],A_{\\sigma(N+1)}],\\ldots,A_{\\sigma(n)}],\n\\end{multline*}\nand so\n\\begin{align*}\n\\|\\hat{F}_n^{\\sigma,\\alpha}\\|_2^2\n\t&= \\sum_{k_1,\\ldots,k_{n-1}}\\left\\|\n\t\t\\sum_{h_1} \\,F_n^\\sigma(k_1\\otimes\\cdots\\otimes k_{n-1}\\otimes h_1\\otimes h_1)\n\t\t\\right\\|_{\\mathfrak{g}_{CM}}^2 \\\\\n\t&\\le \\sum_{k_1,\\ldots,k_{n-1}} \\left( \\sum_{h_1,e_1} \n\t\t|\\langle \\mathcal{A}(h_1,k_i :i\\in I^c),e_1\\rangle\n\t\t_{\\mathfrak{g}_{CM}}|^2 \\right) \\\\\n\t&\\qquad\\qquad\\qquad\\quad\\times\n\t\t\\left( \\sum_{h_1,e_1} \n\t\t\\|[ [[e_1,h_1],A_{\\sigma(N+1)}],\\ldots,A_{\\sigma(n)}]\\|\n\t\t_{\\mathfrak{g}_{CM}}^2 \\right) \\\\\n\t&= \\left(\\sum_{k_i:i\\in I^c,h_1,e_1} |\\langle \\mathcal{A}(h_1,k_i :i\\in I^c),e_1\\rangle\n\t\t_{\\mathfrak{g}_{CM}}|^2 \\right) \\\\\n\t&\\qquad\\qquad\\qquad\\quad \\times \n\t\t\\left( \\sum_{k_i:i\\in I,h_1,e_1} \n\t\t\\|[ [\\ldots[[e_1,h_1],A_{\\sigma(N+1)}],\\ldots],A_{\\sigma(n)}]\\|\n\t\t_{\\mathfrak{g}_{CM}}^2 \\right) \\\\\n\t&\\le \\|[ [ \\ldots[\\cdot,\\cdot],\\ldots],\\cdot]\\|\n\t\t_{(\\mathfrak{g}_{CM}^*)^{\\otimes N-1}\\otimes\\mathfrak{g}_{CM}}^2\n\t\t\\|[ [ \\ldots[\\cdot,\\cdot],\\ldots],\\cdot]\\|\n\t\t_{(\\mathfrak{g}_{CM}^*)^{\\otimes n-N+1}\\otimes\\mathfrak{g}_{CM}}^2.\n\\end{align*}\n \nNow more generally when $q_\\alpha\\ge2$, we may similarly ``separate'' the\npairs of $h_j$'s as above.  More precisely, define\n\\[ \\Phi(b) := \\Phi_\\alpha(b) := \\left\\{  \\begin{array}{ll} \n\tb & \\text{if } b=1,\\ldots, p_\\alpha \\\\\n\t\\lceil\\frac{b-p_\\alpha}{2}\\rceil+p_\\alpha & \\text{if } b=p_\\alpha+1,\\ldots,n\n\\end{array}\\right. .\\]\nLet $N_0=1$, and set\n\\[ \\Omega_1^j := \\{ \\Phi(\\sigma(\\ell)) : \\ell=N_0,\\ldots,j-1\\} \\quad\n\\text{ and } \\quad N_1 := \\min\\{ j>N_0:\n\t\\Phi(\\sigma(j))\\in\\Omega_1^j\\},  \\]\n\\[ \\Omega_2^j := \\{\\Phi(\\sigma(\\ell)):\\ell=N_1,\\ldots,j-1\\} \\quad\n\\text{ and } \\quad N_2 := \\min\\{ j>N_1:\n\t\\Phi(\\sigma(j))\\in\\Omega_2^j\\}. \\]\nSimilarly, we define \n\\[ \\Omega_{2m+1}^j := \\{\\Phi(\\sigma(\\ell)):\\ell=N_{2m},\\ldots,j-1 \\}, \n\t \\]\n\\[ N_{2m+1} := \\min\\left\\{ j>N_{2m}: \\Phi(\\sigma(j))\\in \\bigcup_{i=0}^{m-1}\n\t\\Omega_{2i+1}^{N_{2i+1}} \\cup \\Omega_{2m+1}^j \\right\\}, \\]\n\\[ \\Omega_{2m}^j := \\{\\Phi(\\sigma(\\ell)):\\ell=N_{2m-1},\\ldots,j-1 \\}, \\text{\n\tand}  \n\\]\n\\[ N_{2m} := \\min\\left\\{ j>N_{2m-1}: \\Phi(\\sigma(j))\\in\n\t\\bigcup_{i=1}^{m-1} \\Omega_{2m}^{N_{2m}}\\cup \\Omega_{2m}^j\\right\\}.\n\\]\n\n\nThen there is an $M<q_\\alpha$ such that the sets\n$\\{A_{\\sigma(N_i)},\\ldots,A_{\\sigma(N_{i+1}-1)}\\}_{i=0}^{M-1}$ and\n$\\{A_{\\sigma(N_M)},\\ldots,A_{\\sigma(n)}\\}$ separate the $h_j$'s in the sense that \nno $h_j$ is repeated inside any one of these sets, and moreover the union of\n``even'' sets contains exactly one copy of each $h_j$ and similarly with the \n``odd'' sets. We can write\n\\begin{align*}\nF_n^\\sigma&(k_1\\otimes\\cdots\\otimes k_{p_\\alpha}\\otimes h_1\\otimes\n\t\th_1\\otimes\\cdots\\otimes h_{q_\\alpha}\\otimes h_{q_\\alpha}) \\\\\n\t&= \\sum_{e_1,\\ldots,e_M} \\bigg\\{\n\t\t\\langle e_1,[ [[A_{\\sigma(1)},A_{\\sigma(2)}],\\ldots],A_{\\sigma(N_1-1)}]\\rangle \\\\\n\t&\\qquad \\times\n\t\t\\left(\\prod_{i=1}^{M-1} \\langle e_{i+1},\n\t\t[ [[e_i,A_{\\sigma(N_i)}],\\ldots],A_{\\sigma(N_{i+1}-1)}]\\rangle\\right)\n\t\t[ [[e_M,A_{\\sigma(N_M)}],\\ldots],A_{\\sigma(n)}]\\bigg\\}.\n\\end{align*}\nIf $M$ is even, then\n\\begin{multline*} \n\\mathcal{A} =\n\t\\langle e_1,[[[A_{\\sigma(1)},A_{\\sigma(2)}],\\ldots],A_{\\sigma(N_1-1)}]\\rangle \\\\\n\t\\times \\left(\\prod_{i=1}^{M\/2-1} \\langle e_{2i+1},\n\t[\n\t[[e_{2i},A_{\\sigma(N_{2i})}],\\ldots],A_{\\sigma(N_{2i+1})-1}]\\rangle\\right)\n\t[ [ [e_M, A_{\\sigma(N_M)}],\\ldots], A_{\\sigma(n)}]\n\\end{multline*}\nis a function of $h_1,\\ldots,h_{q_\\alpha},e_1,\\ldots,e_M$ and $k_i$ for $i\\in I$ some subset\nof $\\{1,\\ldots,n\\}$, and \n\\[ \\mathcal{B} = \n\t\\prod_{i=1}^{M\/2}\\langle e_{2i},\n\t\t[[[e_{2i-1},A_{\\sigma(N_{2i-1})}],\\ldots],A_{\\sigma(N_{2i})-1}]\\rangle \\]\nis a function of $h_1,\\ldots,h_{q_\\alpha},e_1,\\ldots,e_M$ and $k_i$ for $i\\in\nI^c$.  Thus,\n\\[ \n\\|\\hat{F}_n^{\\sigma,\\alpha}\\|_2^2\n\t\\le \\left(\\sum_{k_i:i\\in I,h_1,\\ldots,h_{q_\\alpha},e_1,\\ldots,e_M}\n\t\t\t \\|\\mathcal{A}\\|_{\\mathfrak{g}_{CM}}^2\\right)\n\t\t\\left(\\sum_{k_i:i\\in I^c,h_1,\\ldots,h_{q_\\alpha},e_1,\\ldots,e_M}\n\t\t\t|\\mathcal{B}|^2\\right) \\]\nwhich is finite again by Proposition \\ref{p.normest}.\nSimilarly, if $M$ is odd,\n\\begin{multline*} \n\\mathcal{A} = \\langle\n\t\te_1,[[[A_{\\sigma(1)},A_{\\sigma(2)}],\\ldots],A_{\\sigma(N_1-1)}]\\rangle \\\\\n\t\\times \\prod_{i=1}^{(M-1)\/2} \\langle e_{2i+1},\n\t\t[ [[e_{2i},A_{\\sigma(N_{2i})}],\\ldots],A_{\\sigma(N_{2i+1})-1}]\\rangle \n\\end{multline*}\nand\n\\[ \n\\mathcal{B} = \\left(\\prod_{i=1}^{(M-1)\/2}\\langle e_{2i},\n\t\t[ [ [e_{2i-1},A_{\\sigma(N_{2i-1})}],\\ldots],A_{\\sigma(N_{2i})-1}]\\rangle\\right)\n\t\t[ [ [e_M, A_{\\sigma(N_M)}],\\ldots],A_{\\sigma(n)}] \n\\]\nare both functions of $h_1,\\ldots,h_{q_\\alpha},e_1,\\ldots,e_M$ and some $k_i$, and\n\\[ \n\\|\\hat{F}_n^{\\sigma,\\alpha}\\|_2^2\n\t\\le \\left(\\sum_{k_i:i\\in I,h_1,\\ldots,h_{q_\\alpha},e_1,\\ldots,e_M}\n\t\t\t |\\mathcal{A}|_{\\mathfrak{g}_{CM}}^2\\right)\n\t\t\\left(\\sum_{k_i:i\\in I^c,h_1,\\ldots,h_{q_\\alpha},e_1,\\ldots,e_M}\n\t\t\t\\|\\mathcal{B}\\|_{\\mathfrak{g}_{CM}}^2\\right) \\]\nis again finite by Proposition \\ref{p.normest} and this completes the proof.\n\\end{proof}\n\\iffalse\nif $M$ is even, get odd number of sets\n\\[ \n\t\\{A_{\\sigma(N_0)},\\ldots,A_{\\sigma(N_1-1)},\n\tA_{\\sigma(N_2)},\\ldots,A_{\\sigma(N_3-1)},\\ldots,  \n\tA_{\\sigma(N_M)},\\ldots,A_{\\sigma(n)}\\} \n\\]\n\\[ \\{A_{\\sigma(N_1)},\\ldots,A_{\\sigma(N_2-1)},\n\tA_{\\sigma(N_3)},\\ldots,A_{\\sigma(N_4-1)},\\ldots,  \n\tA_{\\sigma(N_{M-1})},\\ldots,A_{\\sigma(N_M)-1}\\} \\]\nif $M$ is odd, get even number of sets\n\\[ \\{A_{\\sigma(N_0)},\\ldots,A_{\\sigma(N_1-1)},\n\tA_{\\sigma(N_2)},\\ldots,A_{\\sigma(N_3-1)},\\ldots,  \n\tA_{\\sigma(N_{M-1})},\\ldots,A_{\\sigma(N_M-1)}\\} \\]\n\\[ \\{A_{\\sigma(N_1)},\\ldots,A_{\\sigma(N_2-1)},\n\tA_{\\sigma(N_3)},\\ldots,A_{\\sigma(N_4-1)},\\ldots,  \n\tA_{\\sigma(N_M)},\\ldots,A_{\\sigma(n)}\\} \\]\n\\fi\n\nPropositions \\ref{p.int2} and \\ref{p.HS} now allow us to make the following definition.\n\n\n\\begin{defn}\n\\label{d.BM}\nA {\\em Brownian motion} on $G$ is the continuous $G$-valued process defined by \n\\[g_t = \\sum_{n=1}^r \\sum_{\\sigma\\in\\mathcal{S}_n}\n\t\t\\sum_{m=\\lceil n\/2\\rceil}^n \\frac{c^\\sigma_n }{2^{n-m}}\n\t\t\\sum_{\\alpha\\in\\mathcal{J}_n^m}\n\t\t\\int_{\\Delta_{p_\\alpha}(t)} f_\\alpha(s,t) \\hat{F}_n^{\\sigma,\\alpha}\n\t\t(dB_{s_1}\\otimes\\cdots\\otimes dB_{s_{p_\\alpha}}), \\]\nwhere \n\\[ c_n^\\sigma=(-1)^{e(\\sigma)}\\bigg\/n^2\\begin{bmatrix} n-1 \\\\ e(\\sigma) \n\t\\end{bmatrix}, \\]\n$\\hat{F}_n^{\\sigma,\\alpha}$ is as defined in equation (\\ref{e.Fhat}), \nand $f_\\alpha$ is as defined below equation (\\ref{e.fprime}).\nFor $t>0$, let $\\nu_t=\\mathrm{Law}(g_t)$ be the {\\em heat kernel measure at\ntime $t$}, a probability measure on $G$.\n\\end{defn}\n\n\n\n\\begin{prop}[Finite-dimensional approximations]\n\\label{p.approx}\nFor $G_\\pi$ a finite-dimensional Lie subgroup of\n$G_{CM}$, let $\\pi$ denote orthogonal projection of\n$G_{CM}$ onto\n$G_\\pi$ and let $g^\\pi_t$ be the continuous process on $G_\\pi$\ndefined by\n\\[ g_t^\\pi = \\sum_{n=1}^r \\sum_{\\sigma\\in\\mathcal{S}_n}\n\t\\sum_{m=\\lceil n\/2\\rceil}^n \\frac{c^\\sigma_n }{2^{n-m}}\n\t\t\\sum_{\\alpha\\in\\mathcal{J}_n^m}\n\t\\int_{\\Delta_{p_\\alpha}(t)} f_\\alpha(s,t) \\hat{F}_n^{\\sigma,\\alpha}\n\t(d\\pi B_{s_1}\\otimes\\cdots\\otimes d\\pi B_{s_{p_\\alpha}}), \\]\nwhere the stochastic integrals are defined as in Proposition \\ref{p.bad}.\nThen $g_t^\\pi$ is Brownian motion on $G_\\pi$.  In particular, for\n$G_\\ell=G_{\\pi_\\ell}$ an increasing sequence of finite-dimensional Lie\nsubgroups such that the associated orthogonal projections $\\pi_\\ell$ are\nincreasing to $I_{\\mathfrak{g}_{CM}}$, let\n$g^\\ell_t=g_t^{\\pi_\\ell}$.  Then, for all $t<\\infty$,\n\\begin{equation} \n\\label{e.b}\n\\lim_{\\ell\\rightarrow\\infty}\\mathbb{E}\n\t\\left\\|g^\\ell_t-g_t\\right\\|_\\mathfrak{g}^2 = 0. \n\\end{equation}\n\\end{prop}\n\\begin{proof}\nFirst note that $g_t^\\pi$ solves the Stratonovich equation \n$\\delta g_t^\\pi = L_{g_t^\\pi*}\\delta \\pi B_t$ with $g_0^\\pi=\\mathbf{e}$, see\n\\cite{BenArous89,Castell93,Baudoin04} where $\\langle \\pi B\\rangle_t$ is a\nstandard $\\mathfrak{g}_\\pi$-valued Brownian motion. \nThus, $g_t^\\pi$ is a $G_\\pi$-valued\nBrownian motion.  \n\n\n\nBy equation (\\ref{e.a}) and its preceding discussion,\n\\[ g_t^\\ell = \\sum_{n=1}^r \\sum_{\\sigma\\in\\mathcal{S}_n}\n\t\\sum_{m=\\lceil n\/2\\rceil}^n \\frac{c^\\sigma_n }{2^{n-m}}\n\t\t\\sum_{\\alpha\\in\\mathcal{J}_n^m}\n\t\\sum_{a=0}^{q_\\alpha} b_\\alpha^a t^a J_n^\\ell(\\tilde{f}_\\alpha\n\t\t\\hat{F}_n^{\\sigma,\\alpha})_t, \\]\nand thus, to verify (\\ref{e.b}), it suffices to show that\n\\[ \\lim_{\\ell\\rightarrow\\infty} \\mathbb{E}\\|\\pi_{\\ell}B_t- B_t\\|_\\mathfrak{g}^2 =\n0 \\]\nand\n\\[ \\lim_{\\ell\\rightarrow\\infty} \\mathbb{E} \n\t\\left\\|J_n^\\ell(\\tilde{f}_\\alpha\\hat{F}_n^{\\sigma,\\alpha})_t -\n\tJ_n(\\tilde{f}_\\alpha\\hat{F}_n^{\\sigma,\\alpha})_t\\right\\|^2\n\t = 0, \\]\nfor all $n\\in\\{2,\\ldots,r\\}$, $\\sigma\\in\\mathcal{S}_n$ and\n$\\alpha\\in\\mathcal{J}_n$.\n\nSo let $\\mu_t=\\mathrm{Law}(B_t)$.  Then it is known that, if $V$ is a finite-dimensional subspace\nof $\\mathfrak{g}_{CM}$ and $\\pi_V$ is the orthogonal projection from\n$\\mathfrak{g}_{CM}$ to $V$, then $\\pi_V$ admits a $\\mu_t$-a.s.~unique\nextension to $\\mathfrak{g}$.  Moreover, if $V_n$ is an increasing sequence of\nfinite-dimensional subspaces, then \n\\[ \\lim_{n\\rightarrow\\infty} \\mathbb{E}\\|\\pi_{V_n}B_t- B_t\\|_\\mathfrak{g}^2 = 0;  \\]\nsee for example Section 8.3.3 of \\cite{Stroock2011}.\n\nBy Proposition \\ref{p.HS},\n$\\hat{F}_n^{\\sigma,\\alpha}$ is Hilbert-Schmidt, and recall that\n$\\tilde{f}_\\alpha$ is a deterministic polynomial function in $s$.  Thus\n$J_n^\\ell(\\tilde{f}_\\alpha \\hat{F}_n^{\\sigma,\\alpha})$ and\n$J_n(\\tilde{f}_\\alpha \\hat{F}_n^{\\sigma,\\alpha})$ are \n$\\mathfrak{g}_{CM}$-valued martingales as defined in Proposition \\ref{p.bad},\nand Proposition \\ref{p.bad} gives the desired convergence as well (in\n$\\mathfrak{g}_{CM}$ and thus in $\\mathfrak{g}$).\n\\end{proof}\n\n\\begin{remark}\nIn fact, for each of the stochastic integrals $J_n(\\tilde{f}_\\alpha\n\\hat{F}_n^{\\sigma,\\alpha})$, it is possible to prove the stronger convergence\nthat, for all $p\\in[1,\\infty)$,\n\\[ \\lim_{\\ell\\rightarrow\\infty} \\mathbb{E}\\left[ \\sup_{\\tau\\le t} \n\t\\left\\|J_n^\\ell(\\tilde{f}_\\alpha\\hat{F}_n^{\\sigma,\\alpha})_\\tau -\n\tJ_n(\\tilde{f}_\\alpha\\hat{F}_n^{\\sigma,\\alpha})_\\tau\\right\\|^p\n\t\\right] = 0, \\]\nfor all $n\\in\\{2,\\ldots,r\\}$, $\\sigma\\in\\mathcal{S}_n$ and\n$\\alpha\\in\\mathcal{J}_n$. Again, Proposition \\ref{p.bad} gives the limit for $p=2$ and thus for\n$p\\in[1,2]$.  \nFor $p>2$, Doob's maximal inequality implies it suffices to show that \n\\[ \\lim_{\\ell\\rightarrow\\infty} \\mathbb{E}\n\t\\left\\|J_n^\\ell(\\tilde{f}_\\alpha\\hat{F}_n^{\\sigma,\\alpha})_t -\n\tJ_n(\\tilde{f}_\\alpha\\hat{F}_n^{\\sigma,\\alpha})_t \n\t\\right\\|^p  = 0.  \\] \nSince each $J_n^\\ell(\\tilde{f}_\\alpha \\hat{F}_n^{\\sigma,\\alpha})$ and\n$J_n(\\tilde{f}_\\alpha \\hat{F}_n^{\\sigma,\\alpha})$ has chaos expansion\nterminating at degree $n$, a theorem of Nelson (see Lemma 2 of\n\\cite{Nelson73b} and pp. 216-217 of \\cite{Nelson73c}) implies that,\nfor each $j\\in\\mathbb{N}$, there exists $c_j<\\infty$ such that \n\\[ \\mathbb{E}\\left\\|J_n^\\ell(\\tilde{f}_\\alpha\\hat{F}_n^{\\sigma,\\alpha})_t -\n\t\tJ_n(\\tilde{f}_\\alpha\\hat{F}_n^{\\sigma,\\alpha})_t \n\t\t\\right\\|^{2j}\n\t\\le c_j \\left(\\mathbb{E}\\left\\|\n\t\tJ_n^\\ell(\\tilde{f}_\\alpha\\hat{F}_n^{\\sigma,\\alpha})_t -\n\t\tJ_n(\\tilde{f}_\\alpha\\hat{F}_n^{\\sigma,\\alpha})_t \n\t\t\\right\\|^2\\right)^j. \\]\n\\end{remark}\n\nIn a similar way, one may prove the following convergence for the Brownian\nmotions under right translations by elements of $G_{CM}$.\n\n\\begin{prop}\n\\label{p.rconv}\nFor any $y\\in G_{CM}$, \n\\[ \\lim_{\\ell\\rightarrow\\infty}\n\t\\mathbb{E}\\|g_t^\\ell y -g_t y\\|^2_{\\mathfrak{g}} = 0. \\]\nwhere $g_ty$ is the measurable right group action of $y\\in G_{CM}$ on $g_t\\in\nG$, as in Proposition \\ref{p.gp}.\n\\end{prop}\n\n\\iffalse\n\\begin{proof}\nNote that $g_t=B_t+I_t$ where $B_t$ is $\\mathfrak{g}$-valued Brownian motion\nand $I_t$ is a finite sum of stochastic integrals taking values in\n$\\mathfrak{g}_{CM}$.  Thus,\n\\begin{align*}\ng_t\\cdot h &= g_t+h+\\sum_{k=1}^{r-1} \n\t\t\\sum_{(n,m)\\in\\mathcal{I}_k}  \n\t\ta_{n,m}^k\\mathrm{ad}_{g_t}^{n_1} \\mathrm{ad}_h^{m_1} \\cdots\n\t\t\\mathrm{ad}_{g_t}^{n_k} \\mathrm{ad}_h^{m_k} g_t\n\\end{align*}\nFor any $x,y\\in W$, \n\\begin{align*} \n\\mathrm{ad}_{x+y}&^{n_1} \\mathrm{ad}_h^{m_1} \\cdots\n\t\t\\mathrm{ad}_{x+y}^{n_k} \\mathrm{ad}_h^{m_k} (x+y) \\\\\n\t&= (\\mathrm{ad}_x + \\mathrm{ad}_y)^{n_1} \\mathrm{ad}_h^{m_1} \\cdots\n\t\t(\\mathrm{ad}_{x} +\\mathrm{ad}_{y})^{n_k} \\mathrm{ad}_h^{m_k} (x+y)\n\t\t\\\\\n\t&= \\sum_{\\ell}\n\t\t\\mathrm{ad}_x^{\\ell_1^1}\\mathrm{ad}_{y}^{\\ell_2^1} \\cdots\n\t\t\\mathrm{ad}_x^{\\ell_{n_1-1}^1}\\mathrm{ad}_{y}^{\\ell_{n_1}^1} \\mathrm{ad}_h^{m_1}\\cdots\n\t\t\\mathrm{ad}_x^{\\ell_1^k}\\mathrm{ad}_{y}^{\\ell_2^k} \\cdots\n\t\t\\mathrm{ad}_x^{\\ell_1^k}\\mathrm{ad}_{y}^{\\ell_2^k} \\mathrm{ad}_h^{m_k} (x+y)\n\\end{align*}\n\n\\end{proof}\n\\fi\n\n\n\\begin{remark}\n\tNote that, while the present paper focuses on the case where $\\mu$ is non-degenerate and $B$ is Brownian motion on $G$, the above construction and finite-dimensional approximations would all follow with essentially no modification if one considered instead a Gaussian measure $\\mu$ whose support was, for example, a subspace $\\frak{h}$ of $\\frak{g}$ such that $\\frak{h}$ generates the span of $\\frak{g}$ via the Lie bracket.\n\\end{remark}\n\n\n\\subsection{Quasi-invariance and log Sobolev}\n\\label{s.hki}\n\n\n\nWe are now able to prove Theorem \\ref{t.quasi}, which states that the heat kernel measure $\\nu_t =\n\\mathrm{Law}(g_t)$ is\nquasi-invariant under left and right translation by elements of $G_{CM}$ and\ngives estimates for the Radon-Nikodym derivatives of the ``translated'' measures. Given the results so far, the proof could be given as an application of Theorem 7.3 and Corollary 7.4 of \\cite{DriverGordina2009}. However, we provide here a full proof for the reader's convenience.\n\n{\\it Proof of Theorem \\ref{t.quasi}.}\nFix $t>0$ and $\\pi_0$ an orthogonal projection onto a finite-dimensional\nsubspace $G_0$ of $\\mathfrak{g}_{CM}$.  Let $h\\in G_0$, and\n$\\{\\pi_n\\}_{n=1}^\\infty$ be an increasing sequence of projections such that\n$G_0\\subset \\pi_nG_{CM}$ for all $n$ and $\\pi_n|_{G_{CM}}\\uparrow\nI_{G_{CM}}$.  Let $J^{n,r}_t(h,\\cdot)$ denote the Radon-Nikodym derivative of\n$\\nu_t^n\\circ R_h^{-1}$ with respect to $\\nu_t^n$. Then for each $n$ and for\nany $q\\in[1,\\infty)$, we have the following integrated Harnack inequality\n\\[ \\left(\\int_{G_n} \\left(J^{n,r}_t(h,g)\\right)^q\n\td\\nu_t^n(g)\\right)^{1\/q} \\le\n\\exp\\left(\\frac{(q-1)k}{2(e^{kt}-1)}d_n(e,h)^2\\right)\n\\]\nwhere $k$ is the uniform lower bound on the Ricci curvature as in Proposition\n\\ref{p.Ric} and $d_n$ is Riemannian distance on $G_n$;\nsee for example Theorem 1.6 of \\cite{DriverGordina2009}.\n\n\nBy Proposition \\ref{p.rconv}, we have that for any $f\\in C_b(G)$, the class of bounded continuous\nfunctions on $G$\n\\begin{equation}\n\\label{e.5.7}\n\\begin{split}\n\\int_{G} f(gh) d \\nu_{t}(g)\n\t&= \\mathbb{E}[f(g_th)] \\\\\n\t&= \\lim_{n\\rightarrow\\infty} \\mathbb{E}[f(g_t^nh)]\n\t= \\lim_{n\\rightarrow\\infty}\\int_{G_{n}} (f\\circ i_n)(gh) \\,\nd\\nu_t^{n}(g),\n\\end{split}\n\\end{equation}\nwhere $ i_n:G_n\\rightarrow G$ denotes the inclusion map.  Note that for any $n$\n\\begin{align*}\n\\int_{G_n} |(f\\circ i_n)(gh)|\\,d\\nu_t^n(g)\n\t&= \\int_{G_n} J^{n,r}_t(h,g)|(f\\circ i_n)(g)|d\\nu_t^n(g) \\\\\n\t&\\le \\|f\\circ i_n\\|_{L^{q'}(G_n,\\nu_t^n)} \\exp\\left(\n\t\t\\frac{k(q-1)}{2(e^{kt}-1)}d_n(e,h)^2\n\t\t\\right),\n\\end{align*}\nwhere $q'$ is the conjugate exponent to $q$.\nAllowing $n\\rightarrow\\infty$ in this last inequality yields\n\\begin{equation}\n\\label{e.c}\n\\int_G |f(gh)|\\,d\\nu_t(g)\n\t\\le \\|f\\|_{L^{q'}(G,\\nu_t)} \\exp\\left(\n\t\t\\frac{k(q-1)}{2(e^{kt}-1)}d(e,h)^2\n\t\t\\right),\n\\end{equation}\nby equation \\eqref{e.5.7} and the fact that\nthe length of a path in $G_{CM}$\ncan be approximated by the lengths of paths in the finite-dimensional\nprojections.  That is, for any\n$\\pi_0$ and $\\varphi\\in C^1([0,1],G_{CM})$ with\n$\\varphi(0)=\\mathbf{e}$, there exists an increasing sequence\n$\\{\\pi_n\\}_{n=1}^\\infty$ of orthogonal projections such that $\\pi_0\\subset \\pi_n$,\n$\\pi_n|_{\\mathfrak{g}_{CM}}\\uparrow I_{\\mathfrak{g}_{CM}}$, and \n\\[ \\ell_{CM}(\\varphi) = \\lim_{n\\rightarrow\\infty}\n\t\\ell_{G_{\\pi_n}}(\\pi_n\\circ\\varphi). \\]\nTo see this, let $\\varphi$ be a path in $G_{CM}$.  Then one may show that\n\\begin{align*}\n\\ell_{G_{\\pi_n}}(\\pi_n\\circ\\varphi)\n\t&=\\int_0^1 \\left\\|\\pi_n\\varphi'(s) + \\sum_{\\ell=1}^{r-1}\n\t\tc_\\ell \\mathrm{ad}_{\\pi_n\\varphi(s)}^\\ell \\pi_n\\varphi'(s)\n\t\t\\right\\|_{\\mathfrak{g}_{CM}}\\,ds \n\\end{align*}\nfor appropriate coefficients $c_\\ell$; see for example Section 3 of\n\\cite{Melcher2009}.\nThus, we have proved that \\eqref{e.c} holds for $f\\in C_b(G)$ and $h\\in\n\\cup_{\\pi} G_\\pi$.  As this union is dense in $G$ by\nProposition \\ref{p.length},\ndominated convergence along with the continuity of $d(e,h)$ in $h$ implies\nthat \\eqref{e.c} holds for all $h\\in G_{CM}$.\n\nSince the bounded continuous functions are dense in $L^{q'}(G,\\nu_t)$ (see for\nexample Theorem A.1 of \\cite{Janson1997}), the inequality in (\\ref{e.c}) implies that the\nlinear functional $\\varphi_h:C_b(G)\\rightarrow\\mathbb{R}$ defined by\n\\[ \\varphi_h(f) = \\int_G f(gh)\\,d\\nu_t(g) \\]\nhas a unique extension to an element, still denoted by\n$\\varphi_h$, of $L^{q'}(G,\\nu_t)^*$ which satisfies the bound\n\\[ |\\varphi_h(f)| \\le \\|f\\|_{L^{q'}(G,\\nu_t)}\n\t\\exp\\left(\n\t\t\\frac{k(q-1)}{2(e^{kt}-1)}d(e,h)^2\n\t\t\\right) \\]\nfor all $f\\in L^{q'}(G,\\nu_t)$.  Since $L^{q'}(G,\\nu_t)^*\\cong L^q(G,\\nu_t)$, there\nthen exists a function $J_t^r(h,\\cdot)\\in\nL^q(G,\\nu_t)$ such that\n\\begin{equation}\n\\label{e.d}\n\\varphi_h(f) = \\int_G f(g)J_t^r(h,g)\\,d\\nu_t(g),\n\\end{equation}\nfor all $f\\in L^{q'}(G,\\nu_t)$, and\n\\[ \\|J_t^r(h,\\cdot)\\|_{L^q(G,\\nu_t)}\n\t\\le \\exp\\left(\n\t\t\\frac{k(q-1)}{2(e^{kt}-1)}d(e,h)^2\n\t\t\\right). \\]\n\nNow restricting (\\ref{e.d}) to $f\\in C_b(G)$, we may rewrite this equation as\n\\begin{equation}\n\\label{e.last}\n\\int_G f(g)\\,d\\nu_t(gh^{-1})\n\t= \\int_G f(g) J_t^r(h,g)\\,d\\nu_t(g).\n\\end{equation}\nThen a monotone class argument (again use Theorem A.1 of\n\\cite{Janson1997}) shows that (\\ref{e.last}) is valid for all\nbounded measurable functions $f$ on $G$.  Thus,\n$d(\\nu_t\\circ R_h^{-1})\/d\\nu_t$ exists and is given by $J_t^r(h,\\cdot)$, which is in\n$L^q$ for all $q\\in(1,\\infty)$ and satisfies the desired bound.\n\nA parallel argument gives the\nanalogous result for $d(\\nu_t\\circ L_h^{-1})\/d\\nu_t$.  Alternatively, one\ncould use the right translation invariance just proved along with the facts that $\\nu_t$\ninherits invariance under the inversion map $g\\mapsto g^{-1}$ from its\nfinite-dimensional projections and that $d(e,h^{-1})=d(e,h)$.\n\\hfill$\\square$\n\n\nThe following also records the straightforward fact that the heat kernel measure does not charge $G_{CM}$.\n\n\\begin{prop}\nFor all $t>0$, $\\nu_t(G_{CM})=0$.\n\\end{prop}\n\n\\begin{proof}\nThis follows trivially from the fact that $g_t$ is the sum of a Brownian motion $B_t$ on\n$\\mathfrak{g}$ with a finite sequence of stochastic integrals taking values in\n$\\mathfrak{g}_{CM}$.\n\\end{proof}\n\nThus, $G_{CM}$ maintains its role as a dense subspace of $G$ of\nmeasure 0 with respect to the distribution of the ``group Brownian\nmotion''.  \n\n\\begin{defn}\n\\label{d.cyl} \nA function $f:G\\rightarrow\\mathbb{R}$ is said to be a\n{\\it (smooth) cylinder function} if $f=F\\circ\\pi$ for some\nfinite-dimensional projection $\\pi$ and\nsome (smooth) function $F:G_\\pi\\rightarrow\\mathbb{R}$.  Also, $f$ is a \n{\\it cylinder polynomial} if $f=F\\circ\\pi$ \nfor $F$ a polynomial function on $G_\\pi$.\n\\end{defn}\n\n\n\\begin{thm}\n\\label{t.logsob}\nGiven a cylinder polynomial $f$ on $G$, let\n$\\nabla f:G\\rightarrow\\mathfrak{g}_{CM}$ be the gradient of $f$,\nthe unique element of $\\mathfrak{g}_{CM}$ such that\n\\[ \\langle\\nabla f(g), h \\rangle_{\\mathfrak{g}_{CM}} = \\tilde{h}f(g)\n\t:= f'(g)(L_{g*}h_\\mathbf{e}), \\]\nfor all $h\\in\\mathfrak{g}_{CM}$.  \nThen for $k$ as in Proposition \\ref{p.Ric},\n\\[ \\int_G (f^2\\ln f^2)\\,d\\nu_t -\n\t\t\\left(\\int_G f^2\\,d\\nu_t \\right)\\cdot\\ln\\left(\\int_G\n\t\tf^2\\,d\\nu_t\\right)\n\t\\le 2\\frac{1-e^{-kt}}{k} \\int_G \\|\\nabla f\\|_{\\mathfrak{g}_{CM}}^2\n\t\t\\,d\\nu_t. \\]\n\\end{thm}\n\n\\begin{proof}\nFollowing the method of Bakry and Ledoux applied to $G_P$ (see Theorem 2.9 of \n\\cite{Driver96} for the case needed here) shows that\n\\[ \\mathbb{E}\\left[\\left(f^2\\ln f^2\\right)\\left(g^\\pi_t\\right)\\right] \n\t\t- \\mathbb{E}\\left[f^2\\left(g^\\pi_t\\right)\\right] \n\t\t\\ln\\mathbb{E}\\left[f^2\\left(g_t^\\pi\\right)\\right]\n\t\\le 2 \\frac{1 - e^{-k_\\pi t}}{k_\\pi} \\mathbb{E}\\left\\|(\\nabla^\\pi\n\t\tf)\\left(g^\\pi_t\\right)\\right\\|^2_{\\mathfrak{g}_\\pi},\n\\]\nfor $k_\\pi$ as in equation (\\ref{e.pah}).  Since the function $x\\mapsto\n(1-e^{-x})\/x$ is decreasing and $k\\le k_\\pi$ for all finite-dimensional\nprojections $\\pi$, \nthis estimate also holds with $k_\\pi$ replaced with $k$.  Now applying\nProposition \\ref{p.approx} to pass to the limit as $\\pi\\uparrow I$ gives \nthe desired result.\n\\end{proof}\n\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\begin{comment}\nThroughout this paper, the term ``knot\" is used both for a knot and a link. That is, a knot in this paper may have more than one component. In the classical knot theory, knots are treated as combinatorial\nobjects, for example as equivalence classes of polygons in the\nthree-dimensional space or as equivalence classes of diagrams that are related by Reidemeister moves.\nAiming at classifying knots, many knot invariants have been\nintroduced, for instance various polynomials, or integer valued invariants such as the bridge index, the braid index of the genus \\cite{BZ, cromwell2004} .\nWhile the classification of all knots is wide open, many knot families have been classified.\nFor example the 2-bridge knots have been completely classified in the 1950s\nby the work of Schubert~\\cite{schubert}.\nThe latter has stimulated the development of the theory of $3$-manifolds.\n\\end{comment}\n\nIn his famous paper \\cite{milnor}, Milnor showed how a knot invariant may be related to geometric properties of the space curve representing a knot by providing a lower bound of the\ntotal curvature of an embedded closed curve explicitly in terms of its bridge index of the knot the curve represents. In a similar spirit, Langer and Singer~\\cite{LS} raised a question concerning the equilibrium shape (a physical property) of a closed springy wire and the knot invariants of the knot the closed wire represents. Gallotti and Pierre-Louis~\\cite{gallotti} conjectured that the equilibrium shapes within knot types $K$ where braid index $\\qopname\\relax o{braid}(K)$ and\nbridge index $\\qopname\\relax o{bridge}(K)$ coincide, will be the $\\qopname\\relax o{bridge}(K)$-times covered circle. This conjecture has recently been proven in the case of 2-bridge knots~\\cite{GRvdM}. For the sake of convenience, we shall call a knot type  whose bridge index and braid index coincide a {\\em BB knot}.\n\nHeuristically, it is plausible that the equilibrium shape\nof a very thin springy wire should be, if topologically possible,\nvery close to a $k$-times covered circle.\nRecalling the F\\'ary--Milnor inequality~\\cite{milnor},\nthe minimum value would be\n$k=\\qopname\\relax o{bridge}(K)$.\nOn the other hand, a configuration of a wire that passes\n$\\qopname\\relax o{bridge}(K)$-times around a circle constitutes in\nfact a braid presentation which requires that $k\\ge \\qopname\\relax o{braid}(K)$\nwhich by $\\qopname\\relax o{braid}(K)\\ge\\qopname\\relax o{bridge}(K)$ implies $\\qopname\\relax o{bridge}(K)=\\qopname\\relax o{braid}(K)$. The left of Figure \\ref{experiment} shows an experiment of the equilibrium state of the figure eight knot realized by a closed springy wire, which is clearly not close to a multiply covered circle. Notice that the figure eight knot is not a BB knot since its bridge index is 2 and braid index is 3. On the other hand, our numerical simulations\nindicate that an equilibrium state of a BB knot $K$ is\nclose to a $\\qopname\\relax o{bridge}(K)$-times covered circle as shown in Figure \\ref{fig:sims}.\n\nThe class of BB knots might also be of potential interest to scientists looking for potential candidates of knot types that can be constructed via molecular knots.\nIn~\\cite{micheletti} the authors give a list of knots that either have been constructed or could potentially be constructed as different molecular knots. At least in the symmetric cases, we observe that there seems to be a prevalence for BB knots, in particular among knot classes with larger crossing number.\n\nIn this light, it is a natural question to investigate the\nfamily of all knot or link classes $K$ with $\\qopname\\relax o{bridge}(K)=\\qopname\\relax o{braid}(K)$. This is the main focus of this paper.\nAs a first step towards a complete classification (which might or might\nnot be feasible) we searched the database KnotInfo \\cite{knotinfo} and found that there are 182 one component BB knots with crossing number up to $12$ (they are listed in Table~\\ref{table}).\nNext, we consider certain families of classes\n(namely torus knots\/links, 2-bridge knots\/links, Montesinos links,\nConway algebraic knots)\nand identify (infinite) subfamilies of BB knots within. \n\nMoreover, there are a number of interesting questions\nconcerning the number $\\mathcal B_{n}$ of BB knots with crossing number $n$.\nDoes $\\mathcal B_{n}$ grow exponentially?\nWhat about the ratio of $\\mathcal B_{n}$ over the number of \\emph{all} knots\nwith with crossing number $n$~?\n\n\\begin{figure}[!h]\n\\label{experiment}\n\\includegraphics[scale=.18,trim=20 0 180 0,clip]{figure8_1}\n\\quad%\n\\rotatebox[origin=c]{-20}{\\reflectbox{\\includegraphics[scale=.36]{bribra\/4_1.png}}}\n\\caption{Left: An experiment with a springy wire belonging to the figure-eight class (Wire model manufactured by \\textsc{why knots}, Aptos, in 1980, photographed by Bernd Bollwerk, Aachen);\nRight: A  local minimizer produced by numerical simulation featuring nearly the same shape (see~\\cite{BR} for details).\n}\n\\end{figure}\n\n\nBefore we proceed to the next section, we need to clarify a couple of key terms and concepts used in this paper. First,  in what follows we will use the word ``knot'' to indicate both a knot type or a particular knot embedding. What we mean will depend on the context and should be clear to the reader. We will use the term ``knot class'' when we talk about the set of embeddings of a knot type with certain geometric restrictions.\nMoreover, the term ``knot\" in this paper is also used for the commonly used term ``link\" in other literature. That is, a knot in this paper may have more than one component. When there is a need, we shall make this clear by specifying the number of components in the knot being discussed. For example, the BB knots listed in Table \\ref{table} are all knots with one component. Second, in the definition of BB knots, the knots are un-oriented (since orientation plays no role to the physical property of interest that leads to the definition of BB knots). However our approach to this problem requires the use of the braid index defined for oriented knots. We would like to point out the connection between these two definitions to avoid confusion. Let $K$ be an un-oriented knot, $\\vec{K}$ be an oriented knot corresponding to $K$ (that is, each component in $K$ has been assigned an orientation) and $\\qopname\\relax o{braid}(\\vec{K})$ be the braid index of $\\vec{K}$ (as an oriented knot). Then un-oriented braid index $\\qopname\\relax o{braid}(K)$ of $K$, which is what we have used in the above in the definition of BB knots, is defined as the minimum of $\\vec{K}$ where $\\vec{K}$ runs over all possible orientation assignments to the components of $K$. It is known (a rather obvious fact) that $\\qopname\\relax o{braid}(\\vec{K})\\ge \\qopname\\relax o{bridge}(K)$. Thus if $\\qopname\\relax o{braid}(\\vec{K})= \\qopname\\relax o{bridge}(K)$ for some $\\vec{K}$,  then we must have $\\qopname\\relax o{braid}(\\vec{K})= \\qopname\\relax o{braid}(K)$. Of course, if $k$ has only one component then for any choice of orientation $\\qopname\\relax o{braid}(\\vec K)=\\qopname\\relax o{braid} (K)$.\n\n\n\nWe organize the rest of the paper as follows: In Sections \\ref{sec:bend} and \\ref{numerics} we discuss BB knots made of thin springy wires (so called \\emph{elastic knots}) and consider the  bending energy of elastic knots in a numerical experiment. \nIn Section \\ref{knotfamilies} we identify BB knots among several knot families. In Section \\ref{numberofknots} we show that the number of BB knots with a given crossing number grows exponentially as a function of the crossing number\nand state a few questions.\n\n\\section{Bending energy minimizers within knot classes}\\label{sec:bend}\n\nThe aim of this section is to explain how the BB knots appear in an elementary model\nfor the experiment described in the introduction.\nFor convenience, we will present here a simplified version\nof the model of \\emph{elastic knots} which is described in Section~\\ref{numerics}\nbelow.\n\n\\subsection{A variational problem}\n\nNeglecting all other physical forces such as friction, shear, and twist,\nwe will assume that the behavior of that springy wire is only affected\nby the bending energy of its centerline $\\gamma:\\mathbb{R}\/\\mathbb{Z}\\to\\mathbb{R}^{3}$\n\\[ E_{\\mathrm{bend}}(\\gamma)=\\int_{\\gamma}(\\kappa(s))^{2} ds \\]\nwhere $\\kappa(s)$ denotes the curvature of $\\gamma$ at $s$\nwhile $s$ is an arc-length parameter. Without loss of generality\nwe may assume that $\\gamma$ is parametrized by arc-length which gives\n$E_{\\mathrm{bend}}(\\gamma)=\\|\\gamma''\\|_{L^{2}}^{2}$.\n\nOur aim would be to find global minimizers of $E_{\\mathrm{bend}}$ within a given knot class $K$.\nAs $E_{\\mathrm{bend}}$ is not invariant under scaling, we will consider only unit length embeddings\nwhich leads to considering the set\n\\[ \\mathscr C(K) = \\left\\{\\gamma\\in W^{2,2}(\\mathbb{R}\/\\mathbb{Z},\\mathbb{R}^{3}) \\middle|\n|\\gamma'(s)|\\equiv1, \\gamma(0)=0_{\\mathbb{R}^{3}},\\gamma\\in K\\right\\}. \\]\nHere we fix $\\gamma(0)$ just for technical reasons.\n\nWe will always assume that $K$ is a non-trivial tame knot class,\nfor the other cases are uninteresting.\nThe global minimizer of $E_{\\mathrm{bend}}$ within the unknot class is\njust the circle.\nAs wild knots are not $C^{1}$ and $W^{2,2}\\hookrightarrow C^{1,1\/2}$\nby the Sobolev embedding theorem, $\\mathscr C(K)=\\emptyset$ for wild $K$.\n\n\\subsection{Minimizers}\n\nIt turns out that, except for the trivial knot class, there are no such minimizers:\nAny potential minimizer within $K$ would be $C^{1}$ (due to the Sobolev embedding\n$W^{2,2}\\hookrightarrow C^{1,1\/2}$) and embedded (since it is a knot).\nAccording to~\\cite{DEJvR} %\nthere is a $C^{1}$-neighborhood consisting of knotted curves belonging to $K$ as well.\nIn particular, this neighborhood contains a $W^{2,2}$-neighborhood\nwith the same property which demonstrates that $\\gamma$ is not only\na global minimizer of $E_{\\mathrm{bend}}$ within $K$ but also a local minimizer\nof $E_{\\mathrm{bend}}$ with respect to \\emph{all} curves.\nNow, by Langer and Singer's seminal result~\\cite{LS},\nthe circle is the only curve with the latter property.\nThis fact reflects the observation that there are indeed points of\nself-contact present in physical models.\n\nNevertheless we can consider a minimal sequence $(\\gamma_{k})_{k\\in\\mathbf N}$ with respect to $E_{\\mathrm{bend}}$\nin $\\mathscr C(K)$,\n{\\em i.e.}, $\\gamma_{k}\\in\\mathscr C(K)$ for all $k\\in\\mathbf N$ with $E_{\\mathrm{bend}}(\\gamma_{k})\\to\\inf_{\\mathscr C(K)}E_{\\mathrm{bend}}$ as $k\\to\\infty$.\nBy the fact that $E_{\\mathrm{bend}}(\\gamma)=\\|\\gamma''\\|_{L^{2}}^{2}$\nand $\\gamma(0)=0_{\\mathbb{R}^{3}}$ for all $\\gamma\\in\\mathscr C(K)$\nwe deduce that this minimal sequence is uniformly bounded in $W^{2,2}$.\nBeing a reflexive space, bounded sets in $W^{2,2}$\nare weakly compact.\nTherefore we may extract a subsequence converging to some limit curve $\\gamma_{0}$\nwith respect to the weak $W^{2,2}$-topology.\nAny such limit curve belongs to the weak $W^{2,2}$-closure\nof $\\mathscr{C}(K)$ which will be denoted by $\\overline{\\mathscr{C}(K)}$.\n\nPotentially there can be many of those limit curves, depending on the\nminimal sequence.\nAny of these limit curves has the following properties:\nIt is not embedded (due to~\\cite{LS}, unless $K$ is trivial), so it\nbelongs to the weak $W^{2,2}$-\\emph{boundary} of $\\mathscr C(K)$.\nIts bending energy provides\na lower bound on the bending energy of any curve in $\\mathscr C(K)$.\nThis is due to the fact that $E_{\\mathrm{bend}}$ is lower \\emph{semi}continuous with respect to weak $W^{2,2}$-convergence.\n\n\\subsection{The case of BB knots}\n\nRecall that the F\\'ary--Milner inequality~\\cite{milnor} bounds the\ntotal curvature $\\int\\kappa(s)ds$ of a curve belonging to $K$ by $2\\pi \\qopname\\relax o{bridge}(K)$ such that\nwe have for $\\gamma\\in\\mathscr C(K)$\n\\[ 2\\pi \\qopname\\relax o{bridge}(K)<\\int_{\\gamma}\\kappa(s)ds=\\|\\gamma''\\|_{L^{1}}. \\]\nAs $\\gamma\\mapsto{}\\|\\gamma''\\|_{L^{1}}$ is not continuous with respect to weak\n$W^{2,2}$-convergence it is not clear whether this lower bound\nalso holds for $\\overline{\\mathscr C(K)}$.\nIn fact, using the existence of \\emph{alternating} quadrisecants\nestabished by Denne~\\cite{denne}, this has been\nproven~\\cite[Appendix]{GRvdM} for $\\qopname\\relax o{bridge}(K)=2$. In the following\nwe \\emph{assume} that it also holds for higher-order bridge indices.\nInvoking the Cauchy--Schwarz inequality, the limit curve $\\gamma_{0}\\in\\overline{\\mathscr C(K)}$ satisfies\n\\[ 2\\pi \\qopname\\relax o{bridge}(K)\\le\\int_{\\gamma_{0}}\\kappa(s)ds=\\|\\gamma_{0}''\\|_{L^{1}}\n\\le\\|\\gamma_{0}''\\|_{L^{2}} = \\left(E_{\\mathrm{bend}}(\\gamma_{0})%\n\\right)^{1\/2}. \\]\nUsing a braid representation we can construct\n$C^{2}$-smooth (even $C^{\\infty}$-smooth) curves belonging to $K$ inside the standard\n$\\varrho$-torus in $\\mathbb{R}^{3}$, {\\em i.e.}, the uniform neighborhood of\nwidth $\\varrho\\in(0,1)$ of the unit circle,\nsuch that each disk of the $\\varrho$-torus is intersected $\\qopname\\relax o{braid}(K)$-times.\nTranslating and rescaling,\nwe obtain a family $(\\beta_{\\varrho})_{\\varrho\\in(0,1)}\\subset\\mathscr C(K)$ such that \n$\\beta_{\\varrho}$ converges (with respect to the $W^{2,2}$-norm) to a $\\qopname\\relax o{braid}(K)$-times\ncovered circle of length one as $\\varrho\\searrow0$.\nOf course, the latter has bending energy $(2\\pi \\qopname\\relax o{bridge}(K))^{2}$.\nAs $\\gamma_{0}$ is an $E_{\\mathrm{bend}}$-minimizer within $\\overline{\\mathscr C(K)}$,\nwe arrive, for any $\\varrho\\in(0,1)$, at\n\\[ (2\\pi \\qopname\\relax o{bridge}(K))^{2}\\le\\left(\\int_{\\gamma_{0}}\\kappa(s)ds\\right)^{2}=\\|\\gamma_{0}''\\|_{L^{1}}^{2}\n\\le\\|\\gamma_{0}''\\|^2_{L^{2}} = E_{\\mathrm{bend}}(\\gamma_{0}) \\le E_{\\mathrm{bend}}(\\beta_{\\varrho})\n\\xrightarrow{\\varrho\\searrow0} (2\\pi \\qopname\\relax o{braid}(K))^{2}. \\]\nHere the condition $\\qopname\\relax o{braid}(K)=\\qopname\\relax o{bridge}(K)$ comes into play for it implies that\nall terms in the previous line are equal. Equality in the Cauchy--Schwarz inequality implies\nthat the integrand is constant a.e. Therefore the curvature of the minimizer\n$\\gamma_{0}$ must be equal to $2\\pi \\qopname\\relax o{bridge}(K)$ a.e.\n\n\\subsection{Caveat}\n\nIt is important to note that the latter condition does \\emph{not}\nimply that $\\gamma_{0}$ is a $\\qopname\\relax o{bridge}(K)$-times covered circle.\nIndeed, that would be wrong.\nIn case $\\qopname\\relax o{bridge}(K)=2$ one can rigorously prove~\\cite{GRvdM} that\nany $E_{\\mathrm{bend}}$ minimizer $\\gamma_{0}$ within $\\mathscr C(K)$\nconsists of two circles that either coincide or tangentially meet in\nprecisely one point.\nUp to isometries and reparametrization, the set of those minimizers can be\nparametrized by the angle between the two circles.\nIn the general situation we would expect a number of $\\qopname\\relax o{bridge}(K)$ circles\ntangentially meeting in (at least) one point.\n\nHowever, adding a positive thickness restriction to $\\mathscr C(K)$ or a penalty term to $E_{\\mathrm{bend}}$\nacts like a choice criterion that selects the $\\qopname\\relax o{bridge}(K)$-times covered circle from that family of $E_{\\mathrm{bend}}$-minimizers.\nThis will be outlined below in Section~\\ref{numerics}.\n\n\\section{BB knots realized by elastic wires}\\label{numerics}\n\n\n\\subsection{Elastic knots}\n\nIn Section~\\ref{sec:bend} above we considered the problem\nto minimize $E_{\\mathrm{bend}}$ in the class $\\mathscr C(K)$ of curves\nin the knot class~$K$. In order to do so, we looked at a\nminimal sequence of curves and extracted a subsequence\nthat converges to some limit curve $\\gamma_{0}$.\nThe problem is that these limit curves are not unique.\nFor instance, in the case of 2-bridge torus knots ({\\em i.e.}, the knots $T(n,2)$) we\nobtain an entire one-parameter family of minimizers.\nWhich of them is the most ``realistic'' one,\n{\\em i.e.}, one that would be observed in physical experiments?\n\nTo answer this question,\nwe need to incorporate the fact that we are looking at\nphysical ropes that have some very small thickness in our simulation model.\nThis can be done by either imposing a constraint to the space of curves~\\cite{gallotti,vdM:eke3}, \nor by adding a penalty term to the bending energy~\\cite{sossinsky,GRvdM,vdM:meek}.\nWe will discuss the latter approach, {\\em i.e.}, we shall adopt the following model\n\\[ E_{\\vartheta} = E_{\\mathrm{bend}}+\\vartheta\\mathcal R, \\qquad\\vartheta>0, \\]\nwhere $\\mathcal R$ denotes a self-avoiding functional\nsuch as the ropelength, {\\em i.e.}, the quotient of length over thickness~\\cite{Buck,DEJvR,GM99}.\nNow the minimization problem is well defined~\\cite{GRvdM,vdM:meek},\n{\\em i.e.}, for any $\\vartheta>0$ there is a curve $\\gamma_{\\vartheta}\\in\\mathscr C(K)$\nsuch that $E_{vartheta}(\\gamma_{\\vartheta})=\\inf_{\\mathscr C(K)} E_{\\vartheta}$.\nIn particular, for $\\vartheta\\in(0,1)$ we have that\n\\[ \\|\\gamma_{\\vartheta}''\\|_{L^{2}}^{2}\n\\le E_{\\vartheta}(\\gamma_{\\vartheta})\n= \\inf_{\\mathscr C(K)} E_{\\vartheta}\n\\le E_{\\vartheta}(\\gamma_{1})\n\\le E_{\\mathrm{bend}}(\\gamma_{1}) + \\mathcal R(\\gamma_{1}) \\]\nis uniformly bounded, %\nso we can extract a subsequence as detailed in Section~\\ref{sec:bend}.\nThe limit curve $\\gamma_{0}$ is referred to as an \\emph{elastic knot} for the knot class $K$~\\cite{GRvdM,vdM:meek}.\nNote that, unless $K$ is trivial, $\\gamma_{0}$ is not embedded.\n\nThere are other names for related concepts in the literature, namely\n``stiff knot'' (Gallotti and Pierre-Louis~\\cite{gallotti}),\nand ``normal form'' (Sossinsky who even thought of a complete classification of\nknot classes by this concept, see~\\cite{sossinsky} and references therein).\n\nNote that, a priori, we cannot expect either the minimizers\n$(\\gamma_{\\vartheta})_{\\vartheta>0}$ or the elastic knot $\\gamma_{0}$\nto be unique. A posteriori, one can show that the elastic knot\nfor the 2-bridge torus knot classes is (up to isometries) the doubly covered circle~\\cite{GRvdM}.\nThe proof relies on a generalization of the crookedness estimate in Milnor's proof~\\cite{milnor}.\nCurrently there are no rigorous results concerning elastic knots\nfor other knot classes. However, the $\\qopname\\relax o{braid}(K)$-times covered circle\nis a candidate for an elastic knot for any BB class~$K$.\nVice versa, one might speculate that the BB classes\nare the only ones whose elastic knots are several times\ncovered circles.\n\n\n\\subsection{Simulation of elastic knots}\n\nPhysical and numerical simulations related to elastic knots\nhave been carried out so far\nby Avvakumov and Sossinsky~\\cite{sossinsky}, Gallotti and Pierre-Louis~\\cite{gallotti},\n Gerlach et al.~\\cite{GRvdM}, as well as by one of the authors and colleagues recently in~\\cite{BR,BRR}.\nThe recently launched web application \\textsc{knotevolve}~\\cite{knotevolve} allows for carrying out a large variety of new experiments.\n\nNumerically, it is a challenging problem to approximate elastic knots as we face two forces which push the elastic knots in different directions. The experiments carried out in~\\cite{BR} shed some light on the energy landscape.\nFor instance, it is unlikely that the configuration shown in\nFigure~\\ref{experiment} is an elastic knot for the figure-eight class; see also the discussion in~\\cite[Section~6.3]{GiRvdM}.\n\n\nIn Figure~\\ref{fig:sims} we show some simulation results for BB knots with braid index up to 3. To this end, we applied the algorithm introduced in~\\cite{bartels13,BRR,BR,knotevolve}\nto the  regularization parameter $\\vartheta{}=10^{-4}$\nand initial configurations from the Knot Server~\\cite{knotserver}.\nFor the BB knots with braid index 3, these simulation results still show visible deviation from the\nthree-times covered circle, in spite of \nthe relatively small regularization parameter.\n\n\n\n\\newcommand{\\knot}[3][.2]{%\n\\fbox{\\begin{minipage}[b][38mm]{.22\\textwidth}\n\\includegraphics[scale=#1]{bribra\/#2_#3.png}\\par\\vfill\n\\includegraphics[scale=#1]{bribra\/#2_#3f.png}\n\\end{minipage}}\\makebox[0cm][r]{\\raisebox{1ex}{$#2_{#3}$\\ }}}\n\\newcommand{\\sknot}[3][.15]{%\n\\fbox{\\begin{minipage}[b][25mm]{.17\\textwidth}\n\\includegraphics[scale=#1]{bribra\/#2_#3.png}\\par\\vfill\n\\includegraphics[scale=#1]{bribra\/#2_#3f.png}\n\\end{minipage}}\\makebox[0cm][r]{\\raisebox{1ex}{$#2_{#3}$\\ }}}\n\n\n\\begin{figure}%\n\\knot3{1}\\hfill\n\\knot5{1}\\hfill\n\\knot7{1}\\hfill\n\\knot9{1}\\medskip\n\n\\sknot[.19]8{5}\\hfill\n\\sknot8{10}\\hfill\n\\sknot[.19]8{16}\\hfill\n\\sknot[.17]8{17}\\hfill\n\\sknot8{18}\\medskip\n\n\\knot8{19}\\hfill\n\\knot8{20}\\hfill\n\\knot8{21}\\hfill\n\\knot9{16}\n\n\\caption{Approximations of elastic knots for all thirteen BB classes\nwith crossing number at most nine.\nEach curve is shown from top and front view;\ncolors correspond to local curvature.\nThe knots in the first row as well as $8_{19}$ are torus knots.}\\label{fig:sims}\n\n\\end{figure}\n\n\n\n\n\\section{BB knot identification in knot families}\n\\label{knotfamilies}\n\nIn this section we will consider knot families including the torus knots, the 2-bridge knots, the alternating Montesinos knots, the non-alternating Montesinos knots, and the Conway algebraic knots. For the torus knots, the 2-bridge knots and the alternating Montesinos knots  we are able to identify all knots within these families that have equal bridge index and braid index. For the non-alternating Montesinos knots and the Conway algebraic knots, we have some partial results. The results for the torus knots and the 2-bridge knots are known, we decide to state them here for the sake of completeness. We would like to point out  that the  proof for the case of 2-bridge knots is new.\n\n\\subsection{Torus knots} \\label{A}\nEvery chiral pair of torus knots has a representative that can be presented by a pair of positive integers $p$, $q$ such that $p\\ge q\\ge2$ and is denoted by $T(p,q)$, with the number of components in  $T(p,q)$ given by $\\gcd(p,q)$.  It is well known that $\\qopname\\relax o{braid}(T(p,q))=\\qopname\\relax o{bridge}(T(p,q))=q$ \\cite{Mu2,S2} or \\cite{Sch2007} for a more recent proof. That is, every torus knot has equal bridge index and braid index. It is known that the crossing number of $T(p,q)$ is $(q-1)p$ \\cite{Mu2} hence the number of torus knots with a given crossing number $n$ is at most of the order of $n$. In other words, the number of torus knots will not contribute to the exponential growth of knots with equal bridge index and braid index. It is worthwhile for us to point out that when a torus knot has more than one component, it is assumed that all components are assigned parallel orientations. One component BB torus knots with crossing number up to 12 are listed in Table \\ref{table} marked with a superscript $^t$. \n\n\\subsection{ 2-bridge knots} \n\nThis case is easy to deal with, since the only knots that can arise a 2-braids are the $T(n,2)$ torus knots. Thus we have by default the following theorem:\n\n\\begin{theorem}\\label{torustwobridge} \nLet $K=B(\\alpha,\\beta)$ be a 2-bridge knot. Then $K$ is a BB knot if and only if $K$ is a  $T(n,2)$ torus knot.\n\\end{theorem}\n\n\nIn the following we give a second proof of the above theorem, introducing a method that will help us to deal with Montesinos knots in the next subsection and will be used in the proof of Theorem \\ref{exp_thm}. To explain this method we need to describe the family of 2-bridge knots in more detail.\nIt is known that every 2-bridge knot can be represented by an alternating diagram associated with two co-prime positive integers $0<\\beta<\\alpha$ in the following way. A vector $(a_1,a_2,...,a_n)$\nis called a {\\it standard continued fraction decomposition} of $\\frac{\\beta}{\\alpha}$ if $n$ is odd and all $a_i>0$ and\n$$\n\\frac{\\beta}{\\alpha}=\\frac{1}{a_1+\\frac{1}{a_{2}+\\frac{1}{.....\\frac{1}{a_n}}}}. \n$$\nIt may be necessary to allow $a_n=1$ in order to guarantee that the length of the vector $(a_1,a_2,...,a_n)$ is odd. Under these conditions the standard continued fraction expansion of $\\frac{\\beta}{\\alpha}$ is unique. An oriented 2-bridge knot (also called a {\\em 4-plat} or a {\\em rational knot}), denoted by $\\vec{K}=\\vec{B}(\\alpha,\\beta)$, is then presented by the {\\em standard diagram} as  shown in Figure \\ref{2bridgeone} using the vector $(a_1,a_2,...,a_n)$. Furthermore, without loss of generality for a standard diagram we can assign the component corresponding to the long arc at the bottom of Figure \\ref{2bridgeone} the orientation as shown, since 2-bridge knots are known to be invertible. When a 2-bridge knot has two components, there are two choices for the orientation of the other component. Usually the two different orientations of the other component lead to two different oriented 2-bridge knots \\cite{BZ, cromwell2004}  which may have different braid indices. For example, the braid index of the two-bridge link in Figure \\ref{2bridgeone} is ten, however if we re-orient one of the two components the braid index changes to nine, see Theorem \\ref{2bridge_theorem} below.\n\n\\begin{figure}[htb!]\n\\includegraphics[scale=1]{Figures2018\/fig11}\n\\caption{The 2-bridge knot $\\vec{B}(17426,5075)$ given by $(3,2,3,3,1,2,3,4,4)$.}\n\\label{2bridgeone}\n\\end{figure}\n\nSince all crossings corresponding to a given $a_i$ have the same crossing sign under the given orientation, we will define a signed vector $(b_1,b_2,...,b_n)$ where $b_i =\\pm a_i$ with its sign given by the crossing sign of the crossings corresponding to $a_i$. For example, for $\\vec{K}=\\vec{B}(17426,5075)$ with the orientation shown in Figure \\ref{2bridgeone} we obtain the signed vector $(3,2,3,3,-1,-2,-3,4,-4)$. In \\cite{DEHL2018} the following theorem was established:\n\n\\begin{theorem}\\label{2bridge_theorem} \\cite{DEHL2018} \nLet $\\vec{K}=\\vec{B}(\\alpha,\\beta)$ be an oriented 2-bridge link diagram with signed vector  $(b_1,b_2,...,b_{2k+1})$ in the standard form, then its braid index is given by\n\\begin{equation}\\label{2bridgeformula}\n\\textbf{b}(K)=1+\\frac{2+\\mbox{\\rm sign}(b_1)+\\mbox{\\rm sign}(b_{2k+1})}{4}+\\sum_{b_{2j}>0,1\\le j\\le k}\\frac{b_{2j}}{2}+\\sum_{b_{2j+1}<0,0\\le j\\le k}\\frac{|b_{2j+1}|}{2}.\n\\end{equation}\n\\end{theorem}\n\nThe above theorem allows the computation of the braid index of any 2-bridge link using a minimal diagram. The computation of the braid index of an oriented 2-bridge link was first given by Murasugi \\cite{Mu} using a different method depending on a continued fraction expansion of $\\beta\/\\alpha$ using only even integers. That the two methods are equivalent was shown in \\cite{DEHL2018}. The formulation of Theorem \\ref{2bridge_theorem} allows us to prove Theorem \\ref{torustwobridge} in a way that can be generalized to Montesinos knots, see the next subsection.\n\n\n\\begin{proof}\nThe only if part is trivially true since every torus knot has equal bridge index and braid index as we discussed in Subsection \\ref{A}. Now, if $K=B(\\alpha,\\beta)$ is a BB knot, then we must have $\\qopname\\relax o{braid}(K)=2$ hence there exists an oriented version $\\vec{K}=\\vec{B}(\\alpha,\\beta)$ such that $\\qopname\\relax o{braid}(\\vec{K})=2$. By (\\ref{2bridgeformula}) we have \n$$\n2=1+\\frac{2+\\mbox{\\rm sign}(b_1)+\\mbox{\\rm sign}(b_{2k+1})}{4}+\\sum_{b_{2j}>0,1\\le j\\le k}\\frac{b_{2j}}{2}+\\sum_{b_{2j+1}<0,0\\le j\\le k}\\frac{|b_{2j+1}|}{2},\n$$\nwhere  $(b_1,b_2,...,b_{2k+1})$ is the signed vector of $\\vec{K}=\\vec{B}(\\alpha,\\beta)$. Notice that $\\mbox{\\rm sign}(b_1)=\\mbox{\\rm sign}(b_2)$ if $b_2\\not=0$. Furthermore, if $b_3>0$ then $b_2$ is even hence $b_2>1$. A proof of this can be found in \\cite{DEHL2018}, a reader can also prove this directly by considering the Seifert circle decomposition of $\\vec{K}$. We will prove the theorem in two separate cases:  $\\mbox{\\rm sign}(b_1)=1$ and $\\mbox{\\rm sign}(b_1)=-1$. \n\nCase 1. $\\mbox{\\rm sign}(b_1)=1$. If $\\mbox{\\rm sign}(b_{2k+1})=1$, then\n$$\n0=\\sum_{b_{2j}>0,1\\le j\\le k}\\frac{b_{2j}}{2}+\\sum_{b_{2j+1}<0,0\\le j\\le k}\\frac{|b_{2j+1}|}{2}\\ge \\frac{b_{2}}{2}\\ge 0,\n$$\nhence $b_2=0$. So $k=0$ and $K=T(n,2)$ for some integer $n\\ge 2$. On the other hand, if $\\mbox{\\rm sign}(b_{2k+1})=-1$, then \n\\begin{eqnarray*}\n&&1+\\frac{2+\\mbox{\\rm sign}(b_1)+\\mbox{\\rm sign}(b_{2k+1})}{4}+\\sum_{b_{2j}>0,1\\le j\\le k}\\frac{b_{2j}}{2}+\\sum_{b_{2j+1}<0,0\\le j\\le k}\\frac{|b_{2j+1}|}{2}\\\\\n&\\ge &\n2+\\sum_{b_{2j}>0,1\\le j\\le k}\\frac{b_{2j}}{2}+\\sum_{b_{2j+1}<0,0\\le j\\le k-1}\\frac{|b_{2j+1}|}{2}> 2,\n\\end{eqnarray*}\nwhich is a contradiction hence this case is not possible.\n\nCase 2. $\\mbox{\\rm sign}(b_1)=-1$. If $b_2=0$, then \n\\begin{eqnarray*}\n2&=&1+\\frac{2+\\mbox{\\rm sign}(b_1)+\\mbox{\\rm sign}(b_{2k+1})}{4}+\\sum_{b_{2j}>0,1\\le j\\le k}\\frac{b_{2j}}{2}+\\sum_{b_{2j+1}<0,0\\le j\\le k}\\frac{|b_{2j+1}|}{2}\\\\\n&= &\n1+\\frac{|b_{1}|}{2},\n\\end{eqnarray*}\nhence $b_1=-2$ and $K$ is the Hopf link that is the mirror image of $T(2,2)$ discussed in Case 1 if we ignore the orientation. If $b_2\\not=0$, then $b_2<0$ and it is necessary that $b_1=-1$ and $b_3<0$ in this case (again this can be observed by considering the Seifert circle decomposition of $K$). It follows that \n\\begin{eqnarray*}\n2&=&1+\\frac{2+\\mbox{\\rm sign}(b_1)+\\mbox{\\rm sign}(b_{2k+1})}{4}+\\sum_{b_{2j}>0,1\\le j\\le k}\\frac{b_{2j}}{2}+\\sum_{b_{2j+1}<0,0\\le j\\le k}\\frac{|b_{2j+1}|}{2}\\\\\n&\\ge &\n1+\\frac{1+\\mbox{\\rm sign}(b_{2k+1})}{4}+\\frac{1+|b_{3}|}{2}.\n\\end{eqnarray*}\nThus we must have $b_3=-1$ and $\\mbox{\\rm sign}(b_{2k+1})=-1$. However $b_3=-1$ implies either $b_4=0$ or $b_4>0$. Since $b_4>0$ leads to $\\qopname\\relax o{braid}(\\vec{K})>2$, we must have $b_4=0$, hence $k=1$ and the signed vector of $K$ is of the form $(-1,-(n-2),-1)$ for some $n>2$. This is the mirror image of the torus knot $T(n,2)$ discussed in Case 1 above if we ignore the orientation.\n\\end{proof}\n\n\\subsection {Alternating Montesinos knots} We now consider the set of all alternating Montesinos knots, a family that is much larger than the family of 2-bridge knots. In general, a Montesinos knot $K=M(\\beta_1\/\\alpha_1,\\ldots, \\beta_k\/\\alpha_s,\\delta)$ is a knot with a diagram as shown in Figure \\ref{Montesinos}, where each diagram within a topological circle (which is only for the illustration and not part of the diagram) is a rational tangle $A_j$ that corresponds to some rational number $\\beta_j\/\\alpha_j$ with $|\\beta_j\/\\alpha_j|<1$ and $1\\le j\\le s$ for some positive integer $s\\ge 2$, and $\\delta$ is an integer that stands for an arbitrary number of half-twists, see Figure \\ref{Montesinos}.\n\n\\begin{figure}[htb!]\n\\includegraphics[scale=.4]{Figures2018\/fig14}\n\\caption{A diagram depicting a general Montesinos knot with $s$ rational tangles and $\\delta$ horizontal half-twists. The arrows indicate the potential orientation assignments if the knot is to be oriented.}\n\\label{Montesinos}\n\\end{figure}\n\nThe bridge index of a general Montesinos knot (alternating or nonalternating) is known and given by the following theorem.\n\n\\begin{theorem} \\cite{BoZi} \n\\label{bridgeMknot}\nLet $K=M(\\beta_1\/\\alpha_1,\\ldots, \\beta_s\/\\alpha_s,\\delta)$ be a Montesinos knot with $|\\beta_j\/\\alpha_j|<1$, $1\\le j\\le s$, then \n$\\qopname\\relax o{bridge}(K)=s$.\n\\end{theorem} \n\nAn explicit formula for the braid index of any alternating Montesinos knot is a rather new result \\cite{DEHL2018}. The discussion from here to (and including) Theorem \\ref{Montesinos_formula} is modified from \\cite{DEHL2018}, as it is needed in order to understand the concepts and the formulas used in Theorem \\ref{Montesinos_formula}. \n\nLet $\\vec{K}=\\vec{M}(\\beta_1\/\\alpha_1,\\ldots, \\beta_s\/\\alpha_s,\\delta)$ be an oriented Montesinos knot. Following \\cite{DEHL2018}, we will use the following conventions on $\\vec{K}$.\nIf $\\vec{K}$ is alternating then all fractions $\\beta_j\/\\alpha_j$ have the same sign and this is matched by the sign of $\\delta$ representing the $|\\delta|$ half-twists. As in the case of 2-bridge knots the sign of $\\delta$ and the $\\beta_j\/\\alpha_j$ should not be confused with the sign of individual crossings. For example, the signs of the crossings represented by $\\delta$ may not coincide with the sign of $\\delta$. The sign of the crossings represented by $\\delta$ depends on the orientations of the two strings in the $\\delta$-half twists. Since a knot and its mirror image have the same braid index, for alternating Montesinos knots we only need to consider the case  $\\beta_j\/\\alpha_j>0$ for each $j$. \n\n\n\nand that the crossings in the tangle diagrams are as chosen in the standard drawings of 2-bridge knots as shown in Figure \\ref{tangle}. The conclusion we reach will then be applicable to the case $\\beta_j\/\\alpha_j<0$ for each $j$ that mirrors the Montesinos knots discussed below. Furthermore, if $s=2$,  the a Montesinos knot is actually a 2-bridge knot and thus we only need to consider the case $s\\ge 3$. \nFor our purpose, we can always orient the top long strand in a Montesinos knot diagram from right to left as shown in Figure \\ref{Montesinos} since reversing the orientations of all components in a knot does not change its braid index. \n\nWe will use a standard drawing for each rational tangle $A_j$ which is given by the continued fraction of the rational number $\\beta_j\/\\alpha_j$ and contains an odd number of positive entries, exactly like what we did in the case of 2-bridge knots. That is, we assume that $0<\\beta_j<\\alpha_j$ and $\\beta_j\/\\alpha_j$ has a continued fraction decomposition of the form $(a_{1}^j,a^j_2,...,a_{2q_j+1}^j)$. The four strands that entering\/exiting each tangle are marked as NW, NE, SW and SE. One example is shown at the left of Figure \\ref{tangle}. Here we note that $a_{2q_j+1}^j$ is allowed to equal one if needed to ensure that the vector $(a_{1}^j,a^j_2,...,a_{2q_j+1}^j)$ has odd length. We have\n$$\n\\frac{\\beta_j}{\\alpha_j}=\\frac{1}{a_1^j+\\frac{1}{a_2^j+\\frac{1}{.....\\frac{1}{a_{2q_j+1}^j}}}}. \n$$\nThe closure of a rational tangle is obtained by connecting its NW and SW end points by a strand and connecting its NE and SE end points with another strand (as shown at the left side of Figure \\ref{tangle}). This closure is called the denominator $D(A_j)$ of the rational tangle $A_j$. Notice that $D(A_j)$ results in a normal standard diagram of  the oriented 2-bridge knot $\\vec{B}(\\alpha_j, \\beta_j)$ given by the vector $(a_{1}^j,a^j_2,...,a_{2q_j+1}^j)$ (as shown at the right side of Figure \\ref{tangle}). Finally, we define $(b_{1}^j,b^j_2,...,b_{2q_j+1}^j)$ by $|b^j_m|=a^j_m$ with its sign matching the signs of the corresponding crossings in $\\vec{K}$ under the given orientation. Notice that $(b_{1}^j,b^j_2,...,b_{2q_j+1}^j)$ may be different from that defined for 2-bridge knots since the orientations of the strands here are inherited from $\\vec{K}$. We will use the notation $A_j(b_{1}^j,b^j_2,...,b_{2q_j+1}^j)$ to denote the tangle $A_j$ and the signed vector associated with it that is inherited from the orientation of $\\vec{K}$. Notice further that in a standard Montesinos knot diagram, each rational tangle has at the bottom a vertical row of $a_1$ twists corresponding to the condition $|\\beta_j\/\\alpha_j|<1$. (Tangles with $|\\beta_j\/\\alpha_j|\\ge1$ end with a row of horizontal twists on the right, and these twists can be combined via flypes with the $\\delta$ half twists to reduce the tangle to $|\\beta_j\/\\alpha_j| \\mod(1)$.)\n\n\\begin{figure}[htb!]\n\\includegraphics[]{fig15x.pdf}\n\\caption{Left: A standard drawing of the rational tangle $56\/191 = A(3,2,2,3,3)$; Right: The denominator $D(A(3,2,2,3,3))$ is a standard 2-bridge knot diagram of the 2-bridge knot $K(191,56)$.}\n\\label{tangle}\n\\end{figure}\n\nLet us now consider the Seifert circle decomposition of $\\vec{K}$ by first examining how the arcs of Seifert circles entering and exiting each $A_j(b_{1}^j,b^j_2,...,b_{2q_j+1}^j)$ might look. Figure \\ref{decomp} lists all eight possibilities for these arcs. Small Seifert circles within each tangle are not shown in Figure \\ref{decomp}. Observing (from Figure \\ref{tangle}) that the SW and SE strands meet at the last crossing in $b_{1}^j$, therefore if these two strands belong to two different Seifert circles, then they must have parallel orientation. Thus (vi) and (viii) are not possible. Furthermore, since we have assigned the top long arc in the Montesinos knot diagram the orientation from right to left, (iii) is not possible either. We say that $A_j$ is of {\\em Seifert Parity 1} if it decomposes as (i) in Figure \\ref{decomp}, of {\\em Seifert Parity 2} if it decomposes as (ii) or (iv) in Figure \\ref{decomp} and of {\\em Seifert Parity 3} if it decomposes as (v) or (vii) in Figure \\ref{decomp}. Note that the Seifert Parity of a tangle $A_j$ is not a property of the tangle alone, but depends on the structure of the diagram that contains $A_j$. (It should not be confused with the the term {\\em parity of a tangle}, which denotes how the arcs in a tangle are connected.)\n\n\\begin{figure}[htb!]\n\\includegraphics{fig16x}\n\\caption{Of the eight cases listed, (iii), (vi) and (viii) are not possible.}\n\\label{decomp}\n\\end{figure}\n\n\\medskip\n\\begin{remark}\\label{bj_sign}{\\em \nBy assigning the appropriate orientations to the two strands at SW and SE in the bottom portion of Figure \\ref{tangle}, one can relate the sign of $b_1^j$ to the Seifert Parity of $A_j$ as follows: $b_1^j<0$ if $A_j$ is of Seifert Parity 1 or 2 and  \n$b_1^j>0$ if $A_j$ is of Seifert Parity 3.}\n\\end{remark}\n\n\\begin{comment}\nThus, it becomes clear that the Seifert circle decomposition of $\\vec{K}$ contains the following: the Seifert circle(s) that contain the top and bottom long strands in $\\vec{K}$ (called {\\em huge} Seifert circle(s)), the Seifert circles that do not contain these long strands, but contain strands that enter\/exit one or more tangles (these will be called {\\em large} Seifert circles), and Seifert circles that are completely within a tangle (and these will be called {\\em medium} Seifert circles if they share crossings with a large Seifert circle, and {\\em small} Seifert circles otherwise). Notice that small Seifert circles are generated by anti-parallel half twists given by a single entry $b_{i}^j$ of a tangle $A_j$. $\\vec{K}$ can be classified into one of the following three classes. \n\\end{comment}\n\nLet us call a Seifert circle in the Seifert circle decomposition of $\\vec{K}$ a {\\em long} Seifert circle if it contains either the top long strand or\/and the bottom long strand. $\\vec{K}$ can be partitioned into the following three classes. \n\nClass M1. The top long strand and the bottom long strand belong to two different long Seifert circles, and the bottom long strand also has the orientation from right to left. Notice that $\\vec{K}$ is of Class M1 if and only if every $A_j$ is of Seifert Parity 1 and all crossings in $\\delta$ (if there are any) have negative signs.\n\nClass M2. The top long strand and the bottom long strand belong to two different long Seifert circles and the bottom long strand has the orientation from left to right. Notice that $\\vec{K}$ is of Class M2 if and only if every $A_j$ is of Seifert Parity 2 (more precisely case (ii) in Figure \\ref{decomp}) and in this case $\\delta=0$.\n\nClass M3. The top long strand and the bottom long strand belong to the same long Seifert circle. Notice that $\\vec{K}$ is of Class M3 if and only if all crossings in $\\delta$ (if there are any) have positive signs and at least one tangle is of Seifert Parity 3 where we consider the horizontal half twists given by $\\delta$ as an additional tangle besides the $A_j$ tangles.\n\n\\noindent\nLet $i=1$, 2 or 3 be the Seifert Parity type of $A_j$ and define \n\n\\begin{eqnarray}\n\\Delta_1(A_j)&=&\\Delta_3(A_j)=\\frac{(-1+\\mbox{\\rm sign}(b^j_{2q_j+1}))}{4}+\\Delta(A_j), \\label{Deltaoneorthree}\\\\\n\\Delta_2(A_j)&=&\\frac{(2+\\mbox{\\rm sign}(b^j_1)+\\mbox{\\rm sign}(b^j_{2q_j+1}))}{4}+\\Delta(A_j),\\label{Deltatwo}\n\\end{eqnarray}\nwhere \n$$\n\\Delta(A_j)=\\sum_{b^j_{2m}>0,1\\le m\\le q_j}b^j_{2m}\/2+\\sum_{b^j_{2m+1}<0,0\\le m\\le q_j}|b^j_{2m+1}|\/2.\n$$\n\nThe braid index for an oriented and alternating Montesinos knot $\\vec{K}$ is  given by the following theorem.\n\n\\begin{theorem}\\label{Montesinos_formula} \\cite{DEHL2018} \nLet $\\vec{K}=\\vec{M}(\\beta_1\/\\alpha_1,\\ldots, \\beta_s\/\\alpha_s,\\delta)=\\vec{M}(A_1,A_2,\\ldots, A_s,\\delta)$ be an oriented and alternating Montesinos knot with a normal standard diagram and the signed vector $(b_{1}^j,b^j_2,...,b_{2q_j+1}^j)$ for $A_j$, then we have\n\\begin{eqnarray*}\n\\qopname\\relax o{braid}(\\vec{K})&=&2+\\sum_{1\\le j\\le s}\\Delta_1(A_j)\\ {\\rm{if}}\\ \\vec{K}\\ {\\rm{belongs\\ to\\ Class\\ M1}};\\\\%\n\\qopname\\relax o{braid}(\\vec{K})&=&1+\\sum_{1\\le j\\le s}\\Delta_2(A_j) \\ {\\rm{if}}\\ \\vec{K}\\ {\\rm{belongs\\ to\\ Class\\ M2}};\\\\%\n\\qopname\\relax o{braid}(\\vec{K})&=&\\Delta_0(\\vec{K})+\\sum_{A_j \\in \\Omega_2}\\Delta_2(A_j)+\\sum_{A_j \\in \\Omega_3}\\Delta_3(A_j) \\ {\\rm{if}}\\ \\vec{K}\\ {\\rm{belongs\\ to\\ Class\\ M3}},%\n\\end{eqnarray*}\nwhere $\\Omega_2$, $\\Omega_3$ are the sets of $A_j$'s that have Seifert Parity 2 and 3 respectively, $\\Delta_0(\\vec{K})=\\eta+\\delta-\\min\\{(\\eta+\\delta)\/2-1,\\delta\\}$ and $\\eta=\\vert \\Omega_3\\vert$. \n\\end{theorem}\n\nUsing Theorem \\ref{Montesinos_formula} we can now identify the alternating Montesinos knots where the bridge index equals the braid index.\n\n\\begin{theorem}\n\\label{braidequalsbridgeMknot}\nLet $ \\vec{K}=\\vec{M}(\\beta_1\/\\alpha_1,\\ldots, \\beta_s\/\\alpha_s,\\delta)$ be an oriented and alternating Montesinos knot. Then $\\qopname\\relax o{braid}( \\vec{K})=\\qopname\\relax o{bridge}( K)=s$ if and only if the following conditions hold:\n\n(i) $\\vec{K}$ is of Class M3.\n\n(ii) $\\eta\\ge \\delta+2$ where $\\eta=\\vert \\Omega_3\\vert$ is the number of tangles $A_j$ with Seifert Parity 3.\n\n(iii) If $A_j$  has Seifert Parity 2 then $A_j=(-1,-s_j,-1)$ for some $s_j\\ge 0$. If $A_j$  has Seifert Parity 3 then $A_j=(s_j)$ for some $s_j> 0$.\n\\end{theorem}\n\n\\begin{proof}\n``$\\Longrightarrow$\": Assume that $\\vec{K}=\\vec{M}(\\beta_1\/\\alpha_1,\\ldots, \\beta_s\/\\alpha_s,\\delta)$ is an oriented alternating Montesinos knot with $\\qopname\\relax o{bridge}(K)=\\qopname\\relax o{braid}(\\vec{K})=s$.\n\nIf $\\vec{K}$ is of Class M1 then every $A_j$ has Seifert Parity 1 and there are at least two $A_j$'s with $\\Delta_1(A_j)=0$ since $\\Delta_1(A_j)\\ge 0$ for each $j$.\nBy Remark \\ref{bj_sign}, $\\mbox{\\rm sign}(b^j_1)=-1$. Thus \n$\\Delta_1(A_j)=0$ is possible only if  $\\mbox{\\rm sign}(b^j_{2q_j+1})=-1$ in (\\ref{Deltaoneorthree}).\nHowever then $\\Delta_1(A_j)\\ge \\sum_{b^j_{2m+1}<0,0\\le m\\le q_j}|b^j_{2m+1}|\/2>0$ and therefore $\\Delta_1(A_j)=0$ is not possible.\n\nSimilarly if $\\vec{K}$ is of Class M2 then there exists at least one tangle $A_j$ with $\\Delta_2(A_j)=0$. This is only possible if $\\mbox{\\rm sign}(b^j_1)=\\mbox{\\rm sign}(b^j_{2q_j+1})=-1$ in (\\ref{Deltatwo}).\nHowever in such a case \\newline\n$\\Delta_2(A_j)\\ge \\sum_{b^j_{2m+1}<0,0\\le m\\le q_j}|b^j_{2m+1}|\/2>0$ therefore $\\Delta_2(A_j)=0$ is not possible either. This proves that $\\vec{K}$ must of Class M3 and therefore all tangles must have Seifert Parity 2 or 3.\n\nLet $\\eta$ be the number of tangles $A_j$ with Seifert Parity 3 and $s-\\eta$ be the number of tangles $A_j$ with Seifert Parity 2. We note that the contribution $\\Delta_2(A_j)$ of a tangle with Seifert Parity 2 to the braid index is always an integer (it is just one less than the braid index of the corresponding two bridge link, see Theorem \\ref{2bridgeformula}) and therefore we know that a tangle of Seifert parity 2 must have $\\Delta_2(A_j)\\ge1$. We have\n$$\ns=\\qopname\\relax o{braid}(\\vec{K})\\ge \\Delta_0(\\vec{K})+s-\\eta +\\sum_{A_j \\in \\Omega_3}\\Delta_3(A_j)  \\ge s+\\delta-\\min\\{(\\eta+\\delta)\/2-1,\\delta\\}.\n$$\nThis implies that $0=\\delta-\\min\\{(\\eta+\\delta)\/2-1,\\delta\\}$, hence $\\eta\\ge \\delta+2$.\nMoreover we must have $\\Delta_2(A_j)=1$ for each $A_j$ with Seifert Parity 2 and $\\Delta_3(A_j)=0$ for each $A_j$ with Seifert Parity 3.\n\nIf $A_j$ has Seifert Parity 2 and $\\Delta_2(A_j)=1$, then $\\mbox{\\rm sign}(b^j_1)=-1$ by Remark \\ref{bj_sign} and equation (\\ref{Deltatwo}) becomes\n$$\n1=\\Delta_2(A_j)=\\frac{|b^j_1|}{2}+\\frac{(1+\\mbox{\\rm sign}(b^j_{2q_j+1}))}{4}+\\sum_{b^j_{2m}>0,1\\le m\\le q_j}b^j_{2m}\/2+\\sum_{b^j_{2m+1}<0,0< m\\le q_j}|b^j_{2m+1}|\/2\\ge \\frac{|b^j_1|}{2}.\n$$\nThus $|b^j_1|\\le 2$. If $b^j_1=-2$ then we must have $\\mbox{\\rm sign}(b^j_{2q_j+1})=-1$, however $1=\\Delta_2(A_j)$ is only possible if $q_j=0$ and $A_j=(-2)$, which can be written as $(-1,0,-1)$ since $(2)$ and $(1,0,1)$ are both continued fraction decompositions of $1\/2$. If $b^j_1=-1$ then $2q_j+1\\ge 3$ and we must have $\\mbox{\\rm sign}(b^j_{2q_j+1})<0$ hence $b^j_{2q_j+1}=-1$ as well. In addition we can see that both $\\mbox{\\rm sign}(b^j_2)$ and $\\mbox{\\rm sign}(b^j_3)$ are negative. Thus we must have  $2q_j+1= 3$, that is, $A_j=(-1,-s_j,-1)$ for some integer $s_j>0$. \n\nIf $A_j$ has Seifert Parity 3, then $\\Delta_3(A_j)=0$. $b^j_1>0$ by Remark \\ref{bj_sign}. If $2q_j+1\\ge 3$ then one can easily check that $\\mbox{\\rm sign}(b^j_2)>0$, which would lead to $\\Delta_3(A_j)>0$. Thus $2q_j+1=1$ and $A_j=(s_j)$ for some integer $s_j>0$. \nThus we have shown that conditions (i), (ii) and (iii) are satisfied.\n\n``$\\Longleftarrow$\": This is straight forward by Theorem \\ref{Montesinos_formula}.\n \\end{proof}\n \n\n\\begin{definition}\\label{v_h}{\\em \nLet us call a tangle $A_j$ corresponding to a rational number of the form $\\beta\/\\alpha=1\/s_j$ for some $s_j\\ge 2$ a {\\em vertical} tangle, and a tangle $A_j$ corresponding to a rational number of the form $\\beta\/\\alpha=(s_j+1)\/(s_j+2)$ for some $s_j\\ge 1$ a {\\em horizontal} tangle. \n}\n\\end{definition}\n\nNote that the standard continued fraction decompositions of a vertical and a horizontal tangle are of the form $(s_j)$ and $(1,s_j,1)$ respectively. Using Theorem \\ref{braidequalsbridgeMknot}, we can then identify all BB knots in the set of all un-oriented alternating Montesinos knots as stated in the following theorem.\n \n\\begin{theorem}\n\\label{unorientedM_BBknots}\nLet ${K}={M}(\\beta_1\/\\alpha_1,\\ldots, \\beta_s\/\\alpha_s,\\delta)$ be an un-oriented alternating Montesinos knot. Then $\\qopname\\relax o{braid}({K})=\\qopname\\relax o{bridge}( K)=s$ if and only if $K$ or its mirror image satisfies the following conditions:\n\n(i) every $A_j$ is either a vertical tangle or a horizontal tangle;\n\n(ii) $\\eta\\ge \\delta+2$ where $\\eta$ is the number of $A_j$'s that are vertical.\n\\end{theorem}\n\nThe key observation one needs to make in the proof of Theorem \\ref{unorientedM_BBknots} is that under condition (ii), we can oriented $K$ so that the resulting $\\vec{K}$ belongs to Class M3. We leave the details to the reader.\n \nThe BB alternating Montesinos knots with one component and crossing number up to 12 are listed in Table \\ref{table} indicated by a superscript $^\\dagger$.\n \n\n\\subsection{Non alternating Montesinos knots}\n\nA classification of Montesinos knots (including both the alternating and non-alternating cases) can be found in \\cite{BZ}. A non alternating Montesinos knot will have a minimum diagram when there are no integral twists on the right, that is, $\\delta=0$. We have the following:\n\n\\begin{theorem}\n\\label{braidequalsbridgeMknotnonalt}\nLet $\\vec{K}=\\vec{M}(\\beta_1\/\\alpha_1,\\ldots, \\beta_s\/\\alpha_s,0)$ be an oriented Montesinos knot. If $\\vec{K}$ belongs to Class M3, $A_j=(-1,-s_j,-1)$ for some integer $s_j\\ge 0$ if it has Seifert Parity 2 and $A_j=(s_j)$ for some integer $s_j\\ge 2$ if it has Seifert Parity 3, then $\\qopname\\relax o{bridge}(K)=\\qopname\\relax o{braid}(\\vec{K})$.\n\\end{theorem}\n\n\\begin{proof} Let $\\vec{K}$ be an oriented Montesinos knot satisfying conditions (i) and (ii). \nIt is known that $ \\qopname\\relax o{braid}(\\vec{K})\\le \\gamma(\\vec{K})$ \\cite{Ya}, where $\\gamma(\\vec{K})$ denotes the number of Seifert circles in the Seifert circle decomposition of $\\vec{K}$. Thus we have $\\qopname\\relax o{bridge}(K)\\le \\qopname\\relax o{braid}(\\vec{K})\\le \\gamma(\\vec{K})$, and the result of the theorem follows if we can show that $\\gamma(\\vec{K})=s$ since $s=\\qopname\\relax o{bridge}(K)$ by Theorem \\ref{bridgeMknot}.\nEach tangle $A_j$ with Seifert Parity 2 contributes a Seifert circle (which is contained within the tangle $A_j$).\n Since each Seifert circle that is not completely contained within a tangle (this includes the large Seifert circle) must contain two vertical arcs: one each from a tangle of case (v) and from a tangle of case (vii) as defined in Figure \\ref{decomp} hence each tangle $A_j$ of Seifert Parity 3 also contributes one Seifert circle. It follows that $\\vec{K}$ has exactly $s$ Seifert circles.\n\\end{proof}\n\n\\begin{corollary}\\label{non_un_cor}\nLet ${K}={M}(\\beta_1\/\\alpha_1,\\ldots, \\beta_s\/\\alpha_s,0)$ be a Montesinos knot with $s\\ge 3$ (that is not necessarily alternating). If each $A_j$ or its mirror image is either a vertical or a horizontal tangle (as defined in Definition \\ref{v_h}) and at least two of them are vertical, then $\\qopname\\relax o{bridge}(K)=\\qopname\\relax o{braid}({K})$.\n\\end{corollary}\n\nIf $K$ satisfies the conditions in the corollary, then we can oriented it in a way that it satisfies the conditions of Theorem \\ref{braidequalsbridgeMknotnonalt}. The details are left to the reader.\n\n\\begin{remark}{\\em\nWe note that the proof of Theorem \\ref{braidequalsbridgeMknotnonalt} does not depend on whether $\\vec{K}$ is alternating, thus it also works for an oriented alternating Montesinos knot $\\vec{K}=\\vec{M}(\\beta_1\/\\alpha_1,\\ldots, \\beta_s\/\\alpha_s,\\delta)$ where $\\delta=0$. However this method is not powerful enough to prove Theorem \\ref{braidequalsbridgeMknot} since the Seifert circle decomposition of $\\vec{K}$ contains $s+\\delta$ Seifert circles when $\\vec{K}$ satisfies the conditions of Theorem \\ref{braidequalsbridgeMknot}. That is, $\\vec{K}$ (in its standard diagram form) does not minimize the number of Seifert circles. In \\cite{DEHL2018} we explain an algorithm that converts $\\vec{K}$ to a non alternating and non minimal diagram $D$ that minimizes the number of Seifert circles via so called reduction moves.}\n\\end{remark}\n\nAll non alternating Montesinos knots with one component and up to 12 crossings that satisfy the condition of \nCorollary \\ref{non_un_cor} are listed in Table \\ref{table} and are marked with a superscript $^\\ddagger$. It turns out that this is a complete list of one component BB non alternating Montesinos knots with crossing number up to 12. In fact, we conjecture that this is generally true. See Conjecture \\ref{conjecture1} at the end of the paper.\n\n\\subsection{Knots using Conway basic polyhedra}\n\nIn \\cite{Con67} the concept of Conway basic polyhedra was introduced. Here we will only concentrate on one class of such polyhedra.\nThe Conway basic polyhedra $n^*$ for $n=2k$ and $k\\ge 3$ can be thought of as taking a regular $k$-gon into which we inscribe a second regular $k$-gon such that its vertices touch the midpoints of the sides of the original $k$-gon. Into the smaller $k$-gon we inscribe a third $k$-gon in the same way. The left of Figure \\ref{eightstar} illustrate the case of $k=4$. This will form a knot  diagram $D$ with $n$ crossings if we make the diagram alternating. Moreover the writhe $w(D)=0$ (with proper orientation in case of a link), three Seifert circles and braid index 3. For example $8^*$ is the knot $8_{18}$ (the right of Figure \\ref{eightstar}) and $10^*$ is the knot $10_{123}$. \nWe note that for $n\\not\\equiv 0\\mod(3)$ we obtain a knot diagram and for $n\\equiv 0\\mod(3)$ we obtain a 3-component knot diagram, for example  for $k=3$ we obtain the Borromean rings ($6_2^3$ in Rolfsen notation). Note that when he first introduced these concepts, Conway used two symbols for the six crossing diagram ($k=3$) denoted by $6^{*}$ and $6^{**}$.  The two symbols denote isomorphic graphs that were introduced to make the obtained notation for knots to be nicer. However they are essentially the same basic polyhedron with alternative way to insert tangles. Conway developed a shorthand notation where the two symbols $6^{**}$ and $6^{*}$ are omitted. If there is an initial dot in the beginning of the symbol, then it is $6^{**}$. If there are more than one tangle symbols separated by dots without the basic polyhedron symbol, then it is meant to be $6^{**}$. If there is also a dot in the beginning of the symbol, then it is $6^{*}$.\nIf the tangle substitutions are simple then there is a natural way to view these diagrams as a 3 string braids without increasing the crossing number and we obtain a braid of three strings and bridge number three.\n\n\\begin{figure}[htb!]\n\\includegraphics[scale=.15]{Figures2018\/eightstar.pdf}\\qquad\n\\includegraphics[scale=.15]{Figures2018\/eighteighteen.pdf}\n\\caption{Left: The Conway basic polyhedra $8^*$. Right: The knot $8_{18}$. }\n\\label{eightstar}\n\\end{figure}\n\nUsing the $6^{*}$, $6^{**}$, $8^{*}$ and $10^{*}$ diagrams, we are able to identify all one component BB knots with crossing number up  to 12 that can be constructed using Conway basic polyhedra. They are listed in \nTable \\ref{table} and are marked with a superscript $^*$. We would like to point out that \nwhile the above knot construction using Conway basic polyhedra can be generalized, the generalized construction may not yield a BB knot.\nOne such example is $n^*$ with $n = 3k$ that can be thought of as inscribing four $k$-gons into each other. The simplest example is $9^*$ which forms the knot $9_{40}$. It has a diagram with 4 Seifert circles and its braid index is four, but its bridge number is only three.\n\n\\subsection{ Arborescent knots or Conway algebraic knots.}\n\nConway algebraic knots are a super family of the Montesinos knots \\cite{BoSi, Con67}.\nThe following describes Conway algebraic knots with bridge and braid index three.\nFirst, they are represented in the Conway notation as $[(a_1;b_1)(a_2;b_2)]$ with $|a_i|\\ge 2$ and $|b_i|\\ge 2$ \\cite{Con67}. These Conway algebraic knots may have one or more components, depending on the parity of $a_i$ and $b_i$. For example, if $a_1$, $a_2$ are both even and $b_1$, $b_2$ are both odd then the knot  has only one component. In the case of multiple components we need to assume that the components are  oriented such that $a_i$ and $b_i$ represent parallel oriented half twists. These knots can be either alternating or non-alternating. In the non alternating case they can also be denoted by $[(a_1;b_1)-(a_2;b_2)]$, $[(a_1;b_1)(a_2-1,1;-b_2)]$, $[(a_1-1,1;-b_1+1,-1)(a_2;b_2)]$ or  $[(-a_1+1,-1;b_1-1,1)(a_2;b_2)]$. The braid index of such a Conway algebraic knot is at most $3$ since the standard diagram given by the Conway notation has 3 Seifert circles in its Seifert circle decomposition. Since these are not 2-bridge knots, their bridge index is at least 3. Thus if $K$ is a Conway algebraic knot described here, then we have $\\qopname\\relax o{braid}(K)=\\qopname\\relax o{bridge}(K)=3$.  \n\n\\subsection{Summary of BB knots with crossing number up to 12.}\n\nTable \\ref{table} lists all one component BB knots with crossing number up to 12. The superscriptions are used in the table as follows. $^t$: torus knots; $^\\dagger$: alternating Montesinos knots; $^\\ddagger$: non alternating Montesinos knots; $^*$: knots constructed from Conway basic polyhedra; $^\\flat$: Conway algebraic knots. The symbols used for the knots follow the historical notation for knots with at most 10 crossings \\cite{rolfsen2003knots} and the notation created for the program Knotscape, developed by Morwen and Jim Hoste \\cite{hoste1999knotscape} for knots with more than 10 crossings, where the letters a and n denote alternating and non-alternating knots, respectively. We would like to point out that a few BB knots in Table \\ref{table} belonging to several families discussed above. For example the knot $8_{19}$ is the torus knot $T(4,3)$, the non alternating Montesinos knot corresponding to $[3;3;2;-]$, as well as the knot constructed using the $8^*$ Conway basic polyhedron. Another example is the knot $10_{124}$: it is the torus knot $T(5,3)$ and also the non alternating Montesinos knot $[5;3;2;-]$.\n\n\\begin{center}\n\\begin{longtable}{|llllllll|}\n\\hline\\hline\n $(3_1)^t$&&&&&&&\\\\ \\hline\n $(5_1)^t$&&&&&&&\\\\ \\hline\n $(7_1)^t$&&&&&&&\\\\ \\hline\n $(8_{5})^\\dagger$&$(8_{10})^\\dagger$&$(8_{16})^\\dagger$&$(8_{17})^*$&$(8_{18})^*$&\n $(8_{19})^{t*\\ddagger}$&$(8_{20})^\\ddagger$&$(8_{21})^\\ddagger$\\\\ \\hline\n $(9_1)^t$&$(9_{16})^\\dagger$&&&&&&\\\\ \\hline\n $(10_{46})^\\dagger$&$(10_{47})^\\dagger$&\n $(10_{48})^\\dagger$&$(10_{62})^\\dagger$&$(10_{64})^\\dagger$&$(10_{79})^\\flat$&\n$(10_{82})^*$&$(10_{85})^*$\\\\ \\hline\n$(10_{91})^*$&$(10_{94})^*$&\n$(10_{99})^*$&$(10_{100})^*$&$(10_{104})^*$&$(10_{106})^*$&\n$(10_{109})^*$&$(10_{112})^*$\\\\ \\hline\n$(10_{116})^*$&$(10_{118})^{t*}$&$(10_{123})^*$&$(10_{124})^{t\\ddagger}$&\n$(10_{125})^\\ddagger$&$(10_{126})^\\ddagger$&$(10_{127})^\\ddagger$&$(10_{139})^\\ddagger$\\\\ \\hline\n$(10_{141})^\\ddagger$&$(10_{143})^\\ddagger$&$(10_{148})^\\flat$&$(10_{149})^\\flat$&\n$(10_{152})^\\flat$&$(10_{155})^*$&$(10_{157})^*$&$(10_{159})^*$\\\\ \\hline\n$(10_{161})^*$&&&&&&&\\\\ \\hline\n$(11a_{44})^\\dagger$&$(11a_{47})^\\dagger$&$(11a_{57})^\\dagger$&$(11a_{231})^\\dagger$&\n$(11a_{240})^\\dagger$&$(11a_{263})^\\dagger$&$(11a_{338})^\\dagger$&$(11a_{367})^t$\\\\ \\hline\n$(11n_{71})^\\ddagger$&$(11n_{72})^\\ddagger$&$(11n_{73})^\\ddagger$&$(11n_{74})^\\ddagger$&\n$(11n_{75})^\\ddagger$&$(11n_{76})^\\ddagger$&$(11n_{77})^\\ddagger$&$(11n_{78})^\\ddagger$\\\\ \\hline\n$(11n_{81})^\\ddagger$&&&&&&&\\\\ \\hline\n$(12a_{146})^\\dagger$&$(12a_{167})^\\dagger$&$(12a_{369})^\\dagger$&$(12a_{576})^\\dagger$&\n$(12a_{692})^\\dagger$&$(12a_{801})^\\dagger$&$(12a_{805})^*$&\n$(12a_{815})^\\flat$\\\\ \\hline\n$(12a_{819})^*$&$(12a_{824})^\\flat$&$(12a_{835})^\\dagger$&\n$(12a_{838})^\\dagger$&$(12a_{850})^*$&$(12a_{859})^*$&$(12a_{864})^*$&\n$(12a_{869})^*$\\\\ \\hline\n$(12a_{878})^\\dagger$&$(12a_{898})^*$&$(12a_{909})^*$&\n$(12a_{916})^*$&$(12a_{920})^*$&$(12a_{981})^*$&$(12a_{984})^*$&\n$(12a_{999})^*$\\\\ \\hline\n$(12a_{1002})^*$&$(12a_{1011})^*$&$(12a_{1013})^*$&\n$(12a_{1027})^\\dagger$&$(12a_{1047})^*$&$(12a_{1051})^*$&$(12a_{1114})^*$&\n$(12a_{1120})^*$\\\\ \\hline\n$(12a_{1168})^*$&$(12a_{1176})^*$&$(12a_{1191})^*$&\n$(12a_{1199})^*$&$(12a_{1203})^*$&$(12a_{1209})^*$&$(12a_{1210})^*$&\n$(12a_{1211})^*$\\\\ \\hline\n$(12a_{1212})^*$&$(12a_{1214})^\\dagger$&$(12a_{1215})^*$&\n$(12a_{1218})^*$&$(12a_{1219})^*$&$(12a_{1220})^*$&$(12a_{1221})^*$&\n$(12a_{1222})^*$\\\\ \\hline\n$(12a_{1223})^*$&$(12a_{1225})^*$&$(12a_{1226})^*$&\n$(12a_{1227})^*$&$(12a_{1229})^*$&$(12a_{1230})^*$&$(12a_{1231})^*$&\n$(12a_{1233})^\\dagger$\\\\ \\hline\n$(12a_{1235})^*$&$(12a_{1238})^*$&$(12a_{1246})^*$&\n$(12a_{1248})^*$&$(12a_{1249})^*$&$(12a_{1250})^*$&$(12a_{1253})^*$&\n$(12a_{1254})^*$\\\\ \\hline\n$(12a_{1255})^*$&$(12a_{1258})^*$&$(12a_{1260})^*$&\n$(12a_{1283})^\\dagger$&$(12a_{1288})^\\flat$&$(12n_{113})^\\flat$&$(12n_{114})^\\flat$&\n$(12n_{190})^\\flat$\\\\ \\hline\n$(12n_{191})^\\flat$&$(12n_{233})^\\ddagger$&$(12n_{234})^\\ddagger$&\n$(12n_{235})^\\ddagger$&$(12n_{242})^\\ddagger$&$(12n_{344})^\\flat$&$(12n_{345})^\\flat$&\n$(12n_{417})^*$\\\\ \\hline\n$(12n_{466})^\\ddagger$&$(12n_{467})^\\ddagger$&$(12n_{468})^\\ddagger$&\n$(12n_{472})^\\ddagger$&$(12n_{570})^\\ddagger$&$(12n_{571})^\\ddagger$&$(12n_{574})^\\ddagger$&\n$(12n_{604})^\\flat$\\\\ \\hline\n$(12n_{640})^*$&$(12n_{647})^*$&$(12n_{666})^*$&\n$(12n_{674})^\\flat$&$(12n_{675})^\\flat$&$(12n_{679})^\\flat$&$(12n_{683})^*$&\n$(12n_{684})^*$\\\\ \\hline\n$(12n_{688})^\\flat$&$(12n_{707})^*$&$(12n_{708})^*$&\n$(12n_{709})^*$&$(12n_{721})^\\ddagger$&$(12n_{722})^\\ddagger$&$(12n_{725})^\\ddagger$&\n$(12n_{747})^*$\\\\ \\hline\n$(12n_{748})^*$&$(12n_{749})^*$&$(12n_{750})^*$&\n$(12n_{751})^*$&$(12n_{767})^*$&$(12n_{820})^*$&$(12n_{821})^*$&\n$(12n_{822})^*$\\\\ \\hline\n$(12n_{829})^*$&$(12n_{830})^*$&$(12n_{831})^*$&\n$(12n_{850})^*$&$(12n_{882})^*$&$(12n_{887})^*$&$(12n_{888})^\\flat$&       \n \\\\\\hline\\hline\n \\caption{List of one component BB knots with crossing number up to 12. \\label{table}}\n\\end{longtable}\n\\end{center}\n\n\\vspace{-0.7in}\n\\section{The number of BB knots with a given crossing number}\n\\label{numberofknots}\n\nTable \\ref{table2} summarizes the number of one component BB knots with crossing numbers up to 12. By separating the knots with odd crossing numbers and the knots with with even crossing number, we observe that the number of one component BB knots with a given crossing number $n$ increases as the crossing number increases and it seems that there are more BB knots in the case of an even crossing number greater than six; and (ii) the percentage of BB knots among all knots with the same crossing number decreases as the crossing number increases. What are the general behaviors of these numbers and percentages? More specifically, if we let $\\mathcal{K}_n$ and $\\mathcal{B}_n$ be the numbers of knots and BB knots with crossing number $n$ respectively, then how do $\\mathcal{B}_n$ and $\\mathcal{B}_n\/\\mathcal{K}_n$ behave? While it is \nplausible that $\\lim_{n\\to\\infty}\\mathcal{B}_n\/\\mathcal{K}_n=0$, we do not have a proof of it. However, we can prove that $\\lim_{n\\to\\infty}\\mathcal{B}_n=\\infty$ and does so exponentially. This is stated in Theorem \\ref{exp_thm}.\n\n\n\\begin{table}[!hb]\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \n\\hline  crossing \\# &3&4&5&6&7&8&9&10&11&12\\\\\n\\hline \\# of knots &1&1&2&3& 7&21&49&165&552&2176\\\\\n\\hline \\# of BB knots &1&0&1&0&1&8&2&33&17&119\\\\\n\\hline $\\approx$ percentage &100\\%&0\\%&50\\%&0\\%& 14.3\\%&38.1\\%&4.1\\%&20.0\\%&3.1\\%&5.5\\%\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n \\caption{Number of one component BB knots with respect to the crossing number. \\label{table2}}\n\\end{table}\n\n\\begin{theorem}\\label{exp_thm}\nThe number $\\mathcal{B}_n$ grows exponentially with $n$. Moreover the number of one component BB knots with crossing number $n$ also grows exponentially with $n$.\n\\end{theorem}\n\\begin{proof}\nIn fact, we will prove that the number of BB knots with braid index 3 grows exponentially with the crossing number $n$. Assume that we only generate alternating 3-braids. Then a word representing such a braid can be generated by two symbols $\\sigma_1$ and $\\sigma_2^{-1}$. Using these two symbols there are $2^n$ words of length $n$ each of which representing an alternating braid with $n$ crossings.\nIf we close such a braid we obtain an alternating knot or link diagram with $n$ crossings and at most 3 components.\nNow we need to estimate how many of these diagrams represent distinct knots of $n$ crossings. There are several points we need to consider:\n\n(i) If $w$ is a braid word then any cyclic permutation of this word will yield the same link. Thus up to cyclic permutation there are at least $\\frac{2^n}{n}$ different words.\n\n(ii) Switching between $\\sigma_1$ and $\\sigma_2^{-1}$ in a braid word will result in the same knot. Thus up to cyclic permutation and exchange of $\\sigma_1$ and $\\sigma_2^{-1}$ there are $\\frac{2^{n-1}}{n}$ different words (at least).\n\n(iii) The diagram generated by the braid closure is a reduced alternating knot diagram if $\\sigma_1$ and $\\sigma_2^{-1}$ both occur in the word more than once. \n\n\n(iv) Using the classification theorem in \\cite{Bir93}, we see that a knot diagram obtained by closing a braid satisfying condition (iii) above admits flypes only if the braid is of the form $\\sigma_1^u \\sigma_2^{-1}\\sigma_1^z\\sigma_2^{-v}$, $\\sigma_2^{-u}\\sigma_1\\sigma_2^{-z}\\sigma_1^{v}$, $\\sigma_1^u\\sigma_2^{-v}\\sigma_1^z\\sigma_2^{-1}$ or  $\\sigma_2^{-u}\\sigma_1^v\\sigma_2^{-z}\\sigma_1^{1}$  for some positive integers $u$, $z$, and $v$. We note that $\\sigma_1^u \\sigma_2^{-1}\\sigma_1^z\\sigma_2^{-v}$ is flype equivalent to $\\sigma_1^u\\sigma_2^{-v}\\sigma_1^z \\sigma_2^{-1}$ and $\\sigma_2^{-u}\\sigma_1\\sigma_2^{-z}\\sigma_1^{v}$ is flype equivalent to $\\sigma_2^{-u}\\sigma_1^v\\sigma_2^{-z}\\sigma_1^{1}$.\n\nIt is easy to see that the number of braids satisfying condition (iii) and also admitting flypes is bounded above by $4\\cdot {{n-2}\\choose{2}}<n^3$. Furthermore, the number of words up to cyclic permutation and exchange of $\\sigma_1$ and $\\sigma_2^{-1}$ with at least two each of $\\sigma_1$ and $\\sigma_2^{-1}$ in the word is at least $2^{n-5}\/n$. The alternating knots obtained by closing braids satisfying condition (iii) and that do not admit flypes are all distinct, thus there exists a constant $c>0$ such that when $n$ is large enough the number of such knots is bounded below by $2^{n-5}\/n-n^3> 2^{cn}$. \nNext, we need to estimate how many of these 3 braids are 2-bridge knots using Theorem \\ref{2bridge_theorem}. Using a case by case analysis similar to the method used in the proof of Theorem \\ref{torustwobridge} we can list all vectors of 2-bridge knots with braid index three. If we represent the these 2-bridge knots with an odd length vector using positive integers than we can show that these vector are of the following form:\n$211$, $a12$, $a2b$ and $3a1$  for length three vectors and $a11b1$ and $1a3b1$ for length five vectors,\nwhere $a,b$ are positive integers. No vectors of a length greater than five can have a braid index of three. Thus for a fixed crossing number $n$ the number of different 2-bridge knots with braid index three is linearly bounded above by $c_1 n$  some fixed constant $c_1>0$. So, the number of  3-braids with $\\qopname\\relax o{bridge}(K)=\\qopname\\relax o{braid}(K)=3$ will still growth exponentially. Roughly a third of these will be one component knots hence the number of one component BB knots also grows exponentially with $n$.\n\\end{proof}\n\nWe end our paper with the following two questions and a conjecture.\n\n\\begin{question} {\\em %\nIs $\\lim_{n\\rightarrow\\infty}\\mathcal{B}_n\/{{\\mathcal{K}_n}}=0$ and does this limit approach zero exponentially fast?\n}\n\\end{question}\n\n\n\n\\begin{question} {\\em Is it true that $\\mathcal{B}_{2n}>\\mathcal{B}_{2n+1}$ for all $n\\ge 4$? And why?}\n\\end{question}\n\n\\begin{conjecture}\\label{conjecture1}{\\em \nThe non alternating Montesinos knots described in Theorem \\ref{braidequalsbridgeMknotnonalt} (or Corollary \\ref{non_un_cor}) are the only non alternating Montesinos knots that are BB knots. \n}\n\\end{conjecture}\n\n\\subsection*{Acknowledgement}\nPh.~R.\\@ was supported by German Research Foundation\ngrant RE~3930\/1--1. We would like to thank S\\\"oren Bartels for fruitful\ndiscussions on the numerical simulation of elastic knots.\n\n\n\\newcommand{\\href}[2]{#2}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe problem of noise and decoherence plays the important role both for applications and for fundamental\nphysics. Noise parameters for electron transport provide a valuable information, which is not available from average\ncurrent-voltage dependences. This stimulates development of effective methods of noise investigation. In particular, a lot of\nattention is devoted to shot noise\n\\cite{Blanter}, which is related to discreteness of charge transport and yields direct information on carrier charges. \nEfforts of theorists were invested to studying full counting statistics of shot noise, its non-Gaussian\ncharacter \\cite{Lez} and asymmetry (odd moments)  \\cite{Odd,BKN}. They are also objects of intensive\nexperimental investigations\n\\cite{Reul,Rez}. \n\nRecently it was shown that the zero-bias\nanomaly of the incoherent Cooper-pairs current in a Coulomb blockaded Josephson junction is very\nsensitive to shot noise from an independent source and therefore can be used for noise investigation\n\\cite{SN}. Further experimental and theoretical investigations \\cite{exp,SN-T,Heik} have demonstrated\nthat this method especially useful for studying asymmetry of shot noise, which is connected with \nits non-Gaussian character and is very difficult for detection by other methods of noise spectroscopy\n\\cite{Reul,Rez}. In  addition, other methods to use the Josephson junction as a noise probe were discussed.\n Deblock {\\em et al.} \\cite{Delft} suggested to use the quasiparticle current for noise detection. The\nscheme to use the Josephson junction as a threshold detector of electron counting statistics has also been\ninvestigated theoretically and experimentally \\cite{TN,Pek,Grab}.  The scheme exploited the effect of\nnoise on macroscopical quantum tunneling at currents close to the critical current.\n\nThe goal of this paper is to extend the theoretical analysis of the effect of noise on the low-bias part of\nthe $IV$ curve of a mesoscopic tunnel junction. The previous analysis \\cite{SN,exp,SN-T} addressed the\neffect of shot noise on a Coulomb blockaded Josephson junction in the weak-coupling limit, when \n the Josephson coupling energy $E_J$ was small compared to the Coulomb energy $E_c=e^2\/C$. Here $C$ is the\nrelevant capacitance. Shot noise originated from a low current through a parallel junction, and its effect\non the $IV$ curve of the Josephson junction had asymmetry connected with the non-Gaussian character of shot\nnoise. The effect was  insensitive to counting statistics of the noise current since the latter was so low\nthat the single-electron tunneling events were well separated in time and their correlation was not\nessential. On the basis of the fulfilled analysis one could expect  that a Coulomb blockade normal junction\nis also able to effectively probe noise, and other types of noise different from shot noise can be\ninvestigated, but in order to check these expectations an additional analysis was needed. The present paper\nis to give answers to the following questions:\n\\begin{itemize}\n\\item\nCan the zero-bias anomaly of the probing junction be sensitive to the counting statistics of electrons tunneling through\nthe noise source (noise junction) when the current through the noise junction grows?\n\n\\item Is the zero-bias anomaly of the Coulomb blockaded  {\\em normal} tunnel junction also sensitive to noise similarly to\nthe Coulomb blockaded  Josephson junction?\n\n\\item Is the zero-bias anomaly of the Coulomb blockaded  tunnel junction sensitive to other types of noise different from\nshot noise?\n\n\\item Can the Josephson junction in the strong coupling limit $E_J\\gg E_C$ also probe noise?\n\\end{itemize}\n\nThe paper gives positive answers to all these questions. The content of the paper is the following. Section \\ref{sec2}\nreminds the previous analysis \\cite{SN,exp,SN-T} for the Coulomb blockade Josephson junction in the weak coupling limit $E_J\n\\ll E_c$ and extends it on the case of higher noise currents, when the zero-bias anomaly is expected to be sensitive to the\nelectron counting statistics. The analysis confirms this expectation. At high currents the\neffect is essentially different for the cases of strictly periodical sequence of pulses and of random pulses governed\nby the Poissonian statistics. Section \\ref{sec3} considers\nanother type of noise: phase fluctuations in a monochromatic microwave signal, which result in decoherence of the\nsignal. A similar phase noise  is well known in quantum optics. It is shown that the effect of the microwave on the\n$IV$ curve of the Josephson junction gives direct information on the decoherence time. Section \\ref{sec4} addresses the\neffect of shot noise on the zero-bias anomaly of a normal tunnel junction. The latter is also very sensitive to shot\nnoise and phase fluctuations in an AC input, and therefore can be exploited as a noise detector. Section \\ref{sec5}  considers the\nJosephson junction in the strong coupling limit in low-impedance environment. This case was chosen because it allows to exploit its\nduality to the case of the Josephson junction in the weak coupling limit in high-impedance environment, which was considered in the\nprevious sections. But duality is valid mostly for the equilibrium {Nyquist-Johnson noise. For shot noise\nduality does not work, and the effect of noise on the Josephson junction in low impedance environment \n(superconductive regime) is different from that in high impedance environment (Coulomb blockade regime). The\nimportant feature of the low-impedance case is  the perspective to measure the intrinsic tunneling time,\nwhich is discussed in the end of Sec.\n\\ref{sec5}. \n\n\n\\section{Josephson junction, weak coupling limit, shot noise} \\label{sec2}\n\n\\subsection{Without shot noise}\n\n Let us review well known results concerning the\n$IV$ curve without shot noise. We assume that the  Josephson coupling\nenergy $E_J\\cos \\varphi$ ($\\varphi$ is the Cooper-pair phase difference between two banks of the Josephson\njunction) is weak in comparison with the Coulomb energy $E_c =e^2\/2C$, where\n$C$ is the relevant capacitance (see below), and the  Josephson coupling can be considered as a time-dependent\nperturbation. Then the\nGolden Rule gives for the current  of Cooper pairs \\cite{Aver,SZ,IN}:\n\\begin{equation}\nI={\\pi eE_J^2 \\over \\hbar}[P(2eV)-P(-2eV)]~, \n \\label{IP}\\end{equation}\nwhere the function\n\\begin{equation}\nP(E)={1\\over 2\\pi \\hbar}\\int_0^\\infty dt\\, \\left[e^{iEt\/\\hbar}\\left\\langle\ne^{i\\varphi(t_0)}e^{-i\\varphi (t_0-t)}\\right\\rangle+ e^{-iEt\/\\hbar}\\left\\langle\ne^{i\\varphi (t_0-t)} e^{-i\\varphi(t_0)}\\right\\rangle\\right]\n  \\label{PE} \\end{equation}\ncharacterizes the probability to transfer the energy $E>0$ to environment (or to\nabsorb the energy $|E|$ from environment if $E<0$) at the time $t_0$. In many cases (equilibrium and shot noise included) the\naverage phase correlators do not depend on the time $t_0$ after averaging, and Eq. (\\ref{PE}) is reduced to the Fourier component\nof the average phase correlator \\cite{IN}:\n\\begin{equation}\nP(E)={1\\over 2\\pi \\hbar}\\int_{-\\infty}^\\infty dt\\, e^{iEt\/\\hbar}\\left\\langle\ne^{i\\varphi(t)}e^{-i\\varphi (0)}\\right\\rangle~.\n  \\end{equation}\nHowever studying the linear response to the AC input in Sec. \\ref{AClin} we shall need the more\ngeneral expression Eq.~(\\ref{PE}).\n\n\nSince we use the\nperturbation theory with respect to $E_J$ we can calculate phase fluctuations neglecting $E_J$, i.e.,\ntreating the Josephson junction as a capacitor. The crucial assumption in the phase-fluctuation theory (or\n$P(E)$ theory) is that the phase fluctuations are Gaussian \\cite{IN} and \n\\begin{equation}\n\\langle e^{i\\varphi_0(t_0)}e^{-i\\varphi_0 (t_0-t)}\\rangle \\approx e^{J_0(t)} ~,\n  \\label{Gauss}    \\end{equation}\nwhere the phase--phase correlator\n\\begin{eqnarray}\nJ_0(t) =\\langle[ \\varphi_0(t)\n-\\varphi_0(0)]\\varphi_0(0)\\rangle=J_R(t)+ iJ_I(t) \n= 2\\int_{-\\infty}^\\infty {d\\omega \\over \\omega}\n\\frac{\\mbox{Re} Z(\\omega)}{R_Q}\n \\frac{e^{-i\\omega t} -1}{1-e^{-\\beta \\hbar \\omega}}\n     \\label{phaseCor} \\end{eqnarray}\nis a complex function of time. The subscript 0 points out that the phase fluctuation $\\varphi_0$ is determined by the\nequilibrium Johnson-Nyquist noise in the environment, i.e., in the electric circuit with the impedance\n$Z(\\omega)$. Here $R_Q=h\/4e^2=\\pi \\hbar\/2e^2$ is the quantum resistance for Cooper pairs and $\\beta =1\/k_BT$ is the inverse\ntemperature. Then\n\\cite{IN}\n\\begin{equation}\nP(E)={1\\over \\pi \\hbar}\\mbox{Re} \\left\\{ \\int_0^\\infty dt \\exp\\left[ J_0(t) +\n\\frac{iEt}{\\hbar}\\right]\\right\\}\\, , \n    \\label{P-E}    \\end{equation}\nand the current is\n\\begin{eqnarray}\nI=-{2 eE_J^2 \\over \\hbar^2}\\mbox{Im}\\left\\{\\int_0^\\infty dt e^{J_0(t)}\n \\sin \\left(2eVt\\over \\hbar\\right)\\right\\}~.\n \\label{WC}  \\end{eqnarray}\nThe zero-bias conductance is given by\n\\begin{equation}\nG_0=\\left.{dI\\over dV}\\right|_{V\\rightarrow 0}=-{4 e^2E_J^2 \\over\n\\hbar^3}\\mbox{Im}\\left\\{\\int_0^\\infty t\\,dt e^{J_0(t)}\\right\\} \\,.\n          \\label{G-0}\\end{equation}\n\n\n\\begin{figure\n  \\begin{center}\n    \\leavevmode\n    \\includegraphics[width=0.5\\linewidth]{ShotL-1.eps}\n   \n    \\caption{Electric circuit. a) The Josephson junction voltage-biased through the shunt resistance $R$. b) Parallel to\nthe Josephson junction the normal junction with the capacitance $C_s$ is switched on, which produces shot noise affecting the\ncurrent through the Josephson junction. The large shunt capacitance $C_{sh}$ transforms at finite frequencies the current bias\ninto the voltage bias.}\n  \\label{fig1}\n  \\end{center}\n  \\end{figure}\n\n\nThe electric circuit for our analysis is shown in Fig. \\ref{fig1}a. The\nJosephson tunnel junction  with capacitance $C_J$ is voltage-biased via the shunt\nresistor $R$. Thus in our case  $Z(\\omega) =(1\/R +i\\omega C)^{-1}$ with $C=C_J$, and \n\\begin{eqnarray}\nJ_0(t) = 2\\rho\\int_{-\\infty}^\\infty {d\\omega \\over \\omega}\n\\frac{1}{1 +\\omega^2\\tau^2}\n \\frac{e^{-i\\omega t} -1}{1-e^{-\\beta \\hbar \\omega}}~,\n     \\label{phaseCorT} \\end{eqnarray}\nwhere $\\tau=RC$ and $\\rho=R\/R_Q$. At $T \\to 0$\n($t>0$):\n\\begin{eqnarray}\nJ_0(t) =  \\rho \\left[-e^{t\/\\tau}\\mbox{E}_1\\left({t\\over \\tau}\\right)-\ne^{-t\/\\tau}\\mbox{E}_1\\left(-{t\\over \\tau}+i0\\right) \n-2 \\ln{ t \\over \\tau}\n -2\\gamma -i\\pi\\right] \\,,\n     \\label{J-0}     \\end{eqnarray}\n where $\\gamma=0.577$ is the Euler constant, and\n$\\mbox{E}_1(z)=\\int_1^\\infty (e^{-zt}\/t)\\,dt$ is the exponential integral\n\\cite{AS}. The small imaginary correction $+i0$ to the argument of one of the\nexponential integrals is important for analytic continuation of $\\mbox{E}_1(z)$ from real\n$t$ to the complex plane \\cite{AS} (see below). At $T=0$ $P(E)$ vanishes for $E<0$ since it is the probability of the transfer of\nthe energy $|E|$ from the environment to the junction, which is impossible if $T=0$.\n\nFor further analysis we need the expressions for $J_0$ in the limits of short time\n$t \\ll \\tau$:\n\\begin{eqnarray}\nJ_0(t) \\approx  \\rho \\left[{t^2  \\over \\tau^2}\\left(\\ln{t\\over\n\\tau}+\\gamma -{3\\over 2}\\right) -{i\\pi t\\over \\tau}\\right]\\,,\n     \\label{J-0s}     \\end{eqnarray}\nand long time $t \\gg \\tau$:\n\\begin{eqnarray}\nJ_0(t) = - \\rho \\left(2 \\ln{ t \\over \\tau}\n +2\\gamma  +i\\pi\\right) \\, .\n     \\label{J-0l}     \\end{eqnarray}\n\nThe function $P(E)$, as well as the current $I$, which it determines, have been carefully studied and calculated for arbitrary\n$\\rho$ \\cite{Ing}. But for the goals of our analysis it is useful to present a simplified calculation valid only in the high\nimpedance limit  $\\rho \\gg 1$. In  this limit integration in the expression for the conductance $G_0$, Eq. (\\ref{G-0}),\nshould be done over rapidly oscillating functions and it is difficult to see that the integral exactly  vanishes at $T=0$.\nBut it becomes evident  after rotation of the integration path in the plane of complex time, which corresponds to replacing\n$t$ by $-iy$. This rotation transforms the integration over the positive real semiaxis into the integration over the\nnegative imaginary semiaxis. After this transformation, the complex function $J_0(t) \\to J_0(-iy)$ becomes a\npurely real function, and the conductance $G_0$, which is given by the imaginary part of this function, vanishes. This\nis true for any term\n$V^k$ in the expansion of the current $I$ in voltage $V$ as far as the integral over $t$, which determines this\nterm, converges at long time. Using the asymptotic expression Eq. (\\ref{J-0l}) one can see that the integral is\ndivergent if\n$k>2\\rho -2$. So at zero temperature the current $I$ as a function of $V$  does not vanish completely but is given by a\nsmall nonanalytic power-law term, which can be found in the limit of high $\\rho$ from the following simple steepest\ndescent estimation of the integral.\n\nExpecting that the main contribution comes from long times one can rewrite the expression (\\ref{P-E}) for the $P(E)$ function\nusing the asymptotic expression  (\\ref{J-0l}) for $J_0$:\n\\begin{eqnarray}\nP(E)= {1\\over \\pi \\hbar}\\mbox{Re}\\left\\{\\int _0^\\infty dt\\,\\exp \\left[ - \\rho \\left(2 \\ln{ t \\over \\tau}\n +2\\gamma  +i\\pi\\right) +{iE t\\over \\hbar}\\right]\\right\\}~.\n    \\end{eqnarray}\nThe saddle point is determined by\nthe condition of vanishing first derivative of the argument of the exponential function: $\n-{2\\rho \/ t_0}  + {iE \/\\hbar}=0$. This yields $t_0=-2i\\hbar\/E$. Expanding the argument of the exponential\nfunction around the saddle point ($t'=t-t_0$) we receive \n\\begin{eqnarray}\n P(E)= {1\\over \\pi \\hbar}\\mbox{Re}\\left\\{\\exp\\left[\\rho\\left(-2\\gamma -\n2\\ln {2\\rho\\hbar\\over iE\\tau}\n-i\\pi\\right)+2\\rho\\right]\\right\\} \\int_{-\\infty}^\\infty\ndt'\\,\\exp\\left(-{E^2\\over 4\\rho\\hbar^2} t'^2\\right)\\nonumber \\\\\n=\\exp[2\\rho(1-\\gamma)]{\\tau \\over \\sqrt{\\pi\\rho} \\hbar} \\left(E\\tau\n\\over 2\\rho\\hbar\\right)^{2\\rho-1}~.\n              \\end{eqnarray}\nFinally \n\\begin{equation}\nI={\\pi eE_J^2 \\over \\hbar}P(2eV)=\\sqrt{\\pi \\over {\\rho} } \\exp[2\\rho(1-\\gamma)] {eE_J^2 \\tau\\over \\hbar^2}\\left(eV\\tau \\over\n\\rho\\hbar\\right)^{2\\rho-1}~. \n \\label{curr0}\\end{equation}\n\n\nIn summary, if environment provides only the equilibrium noise, the low-bias part of the $IV$ curve is governed by phase\ncorrelations at very long time, which yield in the limit $\\rho \\gg 1$ an extremely small nonanalytic current.\n\n\n\\subsection{With shot noise}\n\nNow we consider the effect of shot noise from an independent source. Parallel to the Josephson junction there is another junction\n(noise junction) with the capacitance $C_s$ connected with an independent\nDC current source (see Fig. \\ref{fig1}b). The resistance of the noise junction is very large compared to the\nshunt resistance $R$. The role of very large capacitance\n$C_{sh}$ is to shortcircuit the large ohmic resistance of the current source for finite frequencies. The\ncurrent\n$I_s$ through the noise junction produces shot noise, which affects the $IV$\ncurve of the Josephson junction. \n\nIn the presence of shot noise the fluctuating phase $\\varphi=\\varphi_0+\\varphi_s$ consists of\ntwo terms: $\\varphi_0$  from Johnson-Nyquist noise, and $\\varphi_s$ from shot noise. \nFor calculation of the shot-noise fluctuations $\\varphi_s$ we assume that\nthe charge transport through the noise junction is a sequence of current\npeaks $\\delta I=\\mbox{sign}(I_s)e\\sum_i\\delta (t-t_i)$, where $t_i$ are random\nmoments of time when an electron crosses the junction \\cite{Blanter}. We neglect \nduration of the tunneling event itself. The positive sign of\n$I_s$ corresponds to the current shown in Fig. \\ref{fig1}b.  Any peak\ngenerates a voltage pulse  at the Josephson junction:\n$V_s(t)=\\mbox{sign}(I_s)(e\/C)\\sum_i \\Theta (t-t_i) e^{-(t-t_i)\/\\tau}$, where\n$\\Theta(t)$ is the step function and $C=C_J+C_s$. The voltage pulses result in phase jumps determined by\nthe Josephson relation $\\hbar \\partial \\varphi_s\/\\partial t=2eV_s$. The sequences of current and voltage\npeaks and phase jumps are shown in Fig. \\ref{fig2}.\n\nCalculating the contribution from shot\nnoise fluctuations $\\varphi_s$ to the phase correlators one should abandon the assumption that noise is Gaussian\n[Eq. (\\ref{Gauss})]. On the\nother hand, we assume that the phase fluctuation\n$\\varphi_s$ is classical and the values of  $\\varphi_s$ at different moments of\ntime commute. Since equilibrium noise and shot noise are uncorrelated,  the\ngeneralization of Eq. (\\ref{WC}) is \n\\begin{eqnarray}\nI=-{2 eE_J^2 \\over \\hbar^2}\\mbox{Im}\\left\\{\\int_0^\\infty dt e^{J_0(t)}\n\\left\\langle \\sin \\left({2eVt\\over \\hbar}+\\Delta\n\\varphi_s\\right)\\right\\rangle\\right\\} .\n \\label{curG} \\end{eqnarray}\n\nThe phase difference between two moments $t_0$ and $t_0-t$, $\\Delta \\varphi_s=\n\\varphi_s(t_0)-\\varphi_s(t_0-t)=\\sum _i \n\\delta \\varphi_s(t,t_0-t_i)$, is a sum of contributions from random current peaks, each of them determined\nby\n\\begin{eqnarray}\n\\delta \\varphi_s(t,\\tilde t) \n=\\mbox{sign}(I_s)\\pi \\rho\\left\\{ \\Theta (\\tilde t)\\left [ 1-\ne^{-(\\tilde t)\/\\tau}\\right] \n- \\Theta (\\tilde t-t)\\left [ 1-\ne^{-(\\tilde t-t)\/\\tau}\\right]\\right\\}~, \n     \\end{eqnarray}\nwhere $\\tilde t=t_0-t_i$. Since phase jumps are not  small\nfor $\\rho \\gg 1$, one cannot use the perturbation theory with respect to them. \nThe interpretation of the expression Eq. (\\ref{curG}) for the current is straightforward: As far as the\nrandom phase difference $\\Delta \\varphi_s=2e\\int^{t_0}_{t_0-t}\nV_s(t')dt'\/\\hbar$ is classical,  it should be simply added to the phase difference $\\varphi_V=2eVt\/\\hbar$\ngenerated by the constant voltage bias $V$.\n\n\\begin{figure\n  \\begin{center}\n    \\leavevmode\n    \\includegraphics[width=0.9\\linewidth]{ShotL-2.eps}\n   \n    \\caption{Current and voltage pulses, phase jumps.}\n  \\label{fig2}\n  \\end{center}\n  \\end{figure}\n\nLet us consider the case of the Poissonian statistics. If one performs observation during a very long\nperiod of time\n$\\tau_\\infty$, the number of pulses during this period is large and close to $N\\sim |I_s| \\tau_\\infty\/e$. \nKeeping in mind the absence of correlation between pulses, the phase correlator after averaging over\nthe long time $\\tau_\\infty$ is\n\\begin{eqnarray}\n \\left\\langle e^{i\\Delta \\varphi_s}\\right\\rangle =\n \\left\\langle e^{i\\sum _j\\delta \\varphi_s(t, t_0-t_j) }\\right\\rangle \n=  \\left\\langle e^{i\\delta \\varphi_s }\\right\\rangle^N \n\\approx \\left(1+ {\\Phi(t)\\over \\tau_\\infty}\\right)^N ~,\n              \\end{eqnarray}\nwhere $\\Phi(t)=\\Phi_c(t)+i\\Phi_s(t)$\nis determined by the integrals over a single phase jump (corresponding to a single current pulse):\n\\begin{eqnarray}\n\\Phi_s(t)  = \\int_{-\\infty}^\\infty d\\tilde\nt \\sin \\delta \\varphi_s(t,\\tilde t)\n= \\tau {I_s\\over |I_s|} \\left\\{{\\pi \\over 2}+ \\mbox{si} \\left[r\\left(1-\ne^{-t\/\\tau} \\right) \\right]  +\\sin r \\left[\\mbox{ci} r -\n\\mbox{ci} \\left(r  e^{-t\/\\tau} \\right)\\right] -\\cos r\n\\left[\\mbox{si} r - \\mbox{si} \\left(r \ne^{-t\/\\tau} \\right)\\right] \\right\\}\n        \\label{sin}      \\end{eqnarray}\nand\n\\begin{eqnarray}\n\\Phi_c(t)=\\int_{-\\infty}^\\infty d\\tilde\nt\n\\left[\\cos \\delta \\varphi_s(t, \\tilde t) -1\\right] \n= \\tau \\left\\{ \\mbox{ci} \\left[r\\left(1-e^{-t\/\\tau} \\right) \\right]+\n\\cos r\n\\left[\\mbox{ci} r -\n\\mbox{ci} \\left(r  e^{-t\/\\tau} \\right)\\right]  \\right.\\nonumber \\\\ \\left.\n+ \\sin r\n\\left[\\mbox{si} r - \\mbox{si} \\left(r \ne^{-t\/\\tau} \\right)\\right]     \n-\\gamma-\\ln \\left[r\\left(1-\ne^{-t\/\\tau} \n\\right) \\right] \\right\\}-t~. \n     \\label{cos}         \\end{eqnarray}\nHere $\\mbox{si}(x)=-\\int_x^\\infty\n\\sin t\\, dt\/t$ and $\\mbox{ci}(x)=-\\int_x^\\infty \\cos t\\, dt\/t$ are sine and cosine\nintegral functions \\cite{AS}, and $r=\\pi \\rho$. \nIn the limit $\\tau_\\infty\\to \\infty$ and $N\\to \\infty$ at $N\/\\tau_\\infty=|I_s|\/e$ the phase correlator is \n\\begin{eqnarray}\n \\left\\langle e^{i\\Delta \\varphi_s}\\right\\rangle=  \n\\exp\\left[{N\\Phi(t)\\over \\tau_\\infty}\\right] =\\exp\\left[{|I_s|\\over e}\\Phi(t)\\right]~.\n      \\label{Pois}        \\end{eqnarray}\nFinally the expression for the current, Eq. (\\ref{curG}), can be written as\n\\begin{eqnarray}\nI=-{2 eE_J^2 \\over \\hbar^2}\\mbox{Im}\\left\\{\\int_0^\\infty dt e^{J_0(t)}\n\\exp \\left({|I_s| \\over e}\\Phi_c \\right) \\sin \\left({2eVt\\over\n\\hbar}+{|I_s| \\over e} \\Phi_s\\right) \\right\\}.\n \\label{curGg} \\end{eqnarray}\n\nIn contrast to the case without shot noise, when the expansion of the current $I$ in a small voltage bias\nstarts from the nonanalytic term $\\propto V^{2\\rho-1}$, in the presence of shot noise\nthe expansion in  $V$ starts with analytic terms:\n\\begin{equation}\nI=I_0 +G_s V+ a V^2+bV^3~,\n    \\label{Vexp}  \\end{equation}\nwhere \n\\begin{eqnarray}\nG_s=-{ 4e^2 E_J^2 \\over \\hbar^3 }\\int_0^\\infty t\\, dt \\mbox{Im}  \\left\\{e^{J_0(t)}\\right\\}[\\left\\langle\\cos\n\\Delta\\varphi_s\\right\\rangle-1]\n               \\end{eqnarray}\nis the shot-noise conductance, \n\\begin{eqnarray}\nI_0=-{ 2e E_J^2 \\over \\hbar^2 } \\int_0^\\infty  dt \\mbox{Im}  \\left\\{e^{J_0(t)}\\right\\}\n\\left\\langle\\sin \\Delta\\varphi_s \\right\\rangle\n      \\label{ratchet}         \\end{eqnarray}\nis the ratchet current, and the parameters \n\\begin{eqnarray}\na={ 4e^3 E_J^2 \\over \\hbar^4}\\int_0^\\infty t^2\\, dt \\mbox{Im}  \\left\\{e^{J_0(t)}\\right\\}\n\\left\\langle\\sin \\Delta\\varphi_s \\right\\rangle\n               \\end{eqnarray}\nand\n\\begin{eqnarray}\nb={ 8 e^4 E_J^2 \\over 3 \\hbar^5} \\int_0^\\infty t^3\\, dt \\mbox{Im}  \\left\\{e^{J_0(t)}\\right\\}[\\left\\langle\\cos\n\\Delta\\varphi_s\\right\\rangle-1]\n               \\end{eqnarray}\ndetermine the curvature of the  conductance-voltage plot and the shift of the conductance minimum. Without\nshot noise all these integrals vanish at $\\rho \\gg 1$ [see the paragraph after Eq. (\\ref{J-0l})].\n\nIn Refs. \\onlinecite{exp,SN-T} the parameters of the analytic expansion of $I(V)$ were calculated for low\ncurrents $|I_s|\\ll e\/\\tau$ through the noise junction, when voltage pulses and phase jumps are well\nseparated in time (see Fig.\n\\ref{fig2}). Then  the relevant phase correlators are proportional to the density\n$|I_s|\/e$ of pulses in time \\cite{SN-T}:\n\\begin{equation}\n\\langle e^{i \\Delta \\varphi_s} \\rangle -1= \\langle \\cos \\Delta \\varphi_s \\rangle-1\n+i\\langle \\sin \\Delta \\varphi_s \\rangle = {|I_s| \\over e}\\Phi(t)={|I_s| \\over e}[\\Phi_c(t)+i\\Phi_s(t)]~.\n\\end{equation}\n The parameters \ndetermined by $\\left\\langle\\cos \\Delta\\varphi_s\\right\\rangle$ correspond to ``even'' effects (the conductance\nis symmetric with respect to voltage inversion $V\\to -V$), which are present also without shot noise. Therefore the\ncontribution from shot noise to these parameters  should be\nadded to the values derived from Gaussian equilibrium noise. In contrast, the parameters determined by\n$\\left\\langle\\sin \\Delta\\varphi_s \\right\\rangle$  correspond to ``odd'' (asymmetric) effects, which are absent for \nequilibrium noise and are related to the non-Gaussian character of shot noise.\n\nIn the high-impedance limit $\\rho \\gg 1$ it is possible to calculate the parameters\nof the $IV$ curve analytically. As one can see below, the most important\ncontributions to the integrals, which determine $G_s$, $a$, and $b$, come from times $t  \\sim \\tau\n\/\\sqrt{\\rho \\ln \\rho}$ short compared to $\\tau $, and one may use the small-argument expansion for the\nJohnson-Nyquist correlator given in Eq. (\\ref{J-0s}). On the other hand, these times are long enough for using\nasymptotic expansions for the sine and the cosine integrals:\n$\\mbox{si}(x) \\sim -\\cos x\/x -\\sin x\/x^2 $, $\\mbox{ci}(x) \\sim \\sin x\/x -\\cos x\/x^2$. Then one can rewrite Eq. (\\ref{sin})  as \n\\begin{eqnarray}\n\\Phi_s(t) \\approx \\tau \\left\\{{\\pi \\over 2}+ {1\\over r} -{\\cos \\left[r\\left(1-\ne^{-t\/\\tau} \\right) \\right] \\over re^{-t\/\\tau}}-{\\cos \\left[r\\left(1-\ne^{-t\/\\tau} \\right) \\right] \\over r(1-e^{-t\/\\tau})} \\right.  \\nonumber \\\\ \\left.\n+{\\sin \\left[r\\left(1-\ne^{-t\/\\tau} \\right) \\right] \\over r^2e^{-2t\/\\tau}}-{\\sin \\left[r\\left(1-\ne^{-t\/\\tau} \\right) \\right] \\over r^2(1-e^{-t\/\\tau})^2} \\right\\}\n\\approx \\tau \\left[{\\pi \\over 2}-{\\tau\\over rt} \\cos {rt\\over \\tau}  -{\\tau^2\\over r^2t^2} \\sin {rt\\over\n\\tau}\n\\right]  ~.\n      \\label{sinAs}      \\end{eqnarray}\nIn Eq. (\\ref{cos}) it is enough to keep only the main asymptotic terms $\\propto 1\/r$:\n\\begin{eqnarray}\n\\Phi_c(t) \\approx \\tau \\left\\{-{t \\over\\tau} \n+{\\sin \\left[r\\left(1-e^{-t\/\\tau} \\right) \\right] \\over re^{-t\/\\tau}}+{\\sin \\left[r\\left(1-e^{-t\/\\tau} \\right) \\right] \\over\nr(1-e^{-t\/\\tau})}\n -\\gamma-\\ln \\left[r\\left(1-e^{-t\/\\tau} \\right) \\right] \\right\\}\\nonumber \\\\\n\\approx \\tau \\left[-{t \\over\\tau} -\\ln {rt\\over \\tau}  -\\gamma\n+{\\tau \\over rt}\\sin {rt\\over \\tau} \\right]~. \n          \\label{cosAs}  \\end{eqnarray}\n\nLet us consider first the parameters connected with ``even'' effects. For low currents $|I_s| \\ll\ne\/\\tau $:\n\\begin{eqnarray}\nG_s\\approx -{ 4e E_J^2 \\tau \\over \\hbar^3 }|I_s|\\int_0^\\infty t\\, dt \\mbox{Im}\n\\left\\{e^{J_0(t)}\\right\\}{\\tau \\over rt}\\sin {rt\\over \\tau}  \\nonumber \\\\ \n\\approx { 4e E_J^2 \\tau^2\\over \\hbar^3 r}|I_s| \\int_0^\\infty  dt\n\\exp \\left(-\\rho{t^2  \\over \\tau^2}\\ln{\\tau \\over\nt}\\right) \\sin ^2 {rt \\over\\tau} \n\\approx  {\\sqrt{2} \\pi e E_J^2 \\tau^3\\over \\hbar^3 r^{3\/2}\\sqrt{\\ln r}}|I_s|\n\\approx {\\pi^{5\/2}E_J^2 C^3 \\over 4\ne^5 \\sqrt{2\\ln \\rho}}\\rho^{3\/2}|I_s|~,\n    \\label{integr}           \\end{eqnarray}\n\\begin{eqnarray}\nb={ 8 e^3 E_J^2\\tau \\over 3 \\hbar^5}|I_s| \\int_0^\\infty t^3\\, dt \\mbox{Im}  \\left\\{e^{J_0(t)}\\right\\}{\\tau \\over rt}\\sin {rt\\over\n\\tau} \n\\nonumber \\\\\n\\approx -{ 8 e^3 E_J^2\\tau^2 \\over 3 \\hbar^5 r}|I_s| \\int_0^\\infty t^2\\, dt \\exp \\left(-\\rho{t^2  \\over \\tau^2}\\ln{\\tau \\over\nt}\\right) \\sin ^2 {rt \\over\\tau} \\approx -{ 2\\pi^2\\sqrt{2} e^3 E_J^2\\tau^5 \\over 3 \\hbar^5 r^{5\/2} (\\ln r)^{3\/2}}|I_s|\n\\approx -{\\pi^2 C^2\\rho\\over 6 e^2\\ln \\rho}G_s ~.\n       \\label{integr-b}        \\end{eqnarray}\nThe main contributions to these integrals come from the last term in the asymptotic expansion Eq.\n(\\ref{cosAs}). The  other terms either vanish (term $\\propto \\gamma$) or are of higher orders (terms $\\propto\nt$ and $\\propto \\ln t$) with respect to $1\/r$. A negative sign of $b$ means that in the center of the\nzero-bias anomaly the curve ``conductance vs. voltage'' has a maximum but not a minimum. However this is\npossible to see only at very low temperatures since finite temperatures give a positive contribution to the\ncurvature parameter $b$.\n\n\nIn a similar way one can calculate the asymmetry integral \n\\begin{eqnarray}\na=-{ 4e^2 E_J^2 \\tau \\over \\hbar^4}I_s\\int_0^\\infty t^2\\, dt \\mbox{Im}  \\left\\{e^{J_0(t)}\\right\\}\n{\\tau^2\\over r^2t^2} \\sin {rt\\over \\tau}={I_s \\over |I_s|}{  C\\over 2e }G_s ~.\n    \\label{integr-a}      \\end{eqnarray}\nHere the main contribution comes from the last term of the asymptotic expansion Eq.  (\\ref{sinAs}). \n\n For the integral in Eq.\n(\\ref{ratchet}), which determines the ratchet current $I_0$, the relevant times are shorter than for the\nother integrals:\n$t\n\\sim \\tau\/\\rho \\sim R_QC$. Therefore one cannot use the asymptotic expansion Eq. (\\ref{sinAs}). Instead one can\nintegrate by parts, and for low noise currents: \n\\begin{eqnarray}\nI_0=-{ 2e E_J^2 \\over \\hbar^2 } {|I_s|\\over e} \\int_0^\\infty  dt \\mbox{Im}  \\left\\{e^{J_0(t)}\\right\\}\n\\Phi_s(t) \\approx { 2 E_J^2 \\over \\hbar^2 }|I_s| \\int_0^\\infty  dt \\Phi_s(t) \\sin {r t\\over \\tau}\n = { 2 E_J^2\\tau \\over \\hbar^2 r } |I_s| \\int_0^\\infty  dt \\frac {d\\Phi_s(t)}{dt}\\cos {r t\\over \\tau}\n\\nonumber \\\\\n\\approx  { 2E_J^2\\tau^2 \\over \\hbar^2 r }I_s \\int_0^\\infty  {dt\\over t} \\cos {r t\\over \\tau}\n\\sin {r t\\over \\tau}={ \\pi E_J^2\\tau^2 \\over 2 \\hbar^2 r }I_s= {\\pi^2 E_J^2 C^2\\over 8e^4 }\\rho I_s\\, .\n     \\label{integr-r}     \\end{eqnarray}\n\n\\subsection{Shot noise from high currents and electron counting statistics} \\label{sec2c}\n\n\nShot noise is non-Gaussian and asymmetric, but in the case of low noise currents its effect does not\ndepend on statistics of electron transport and is determined only by density of current pulses in time.\nEven if pulses were not random and formed a strictly periodical sequence the effect would be the same. In\norder to obtain information on counting statistics from measurements of the $IV$ curve the noise current\nshould be not small compared with the width of the voltage pulse $\n\\sim \\tau$. One could expect \\cite{SN-T} that in the limit of high noise current $I_s$ discreteness of the\nelectron transport through the junction would be less and less essential and eventually the effect of the\ncurrent\n$I_s$ would be reduced to the constant voltage drop $V_s =I_s  R$ on the shunt resistor $R$ resulting\nin a trivial shift of the voltage applied to the Josephson junction. If it were the case, the long-time asymptote for\nthe phase correlator would be\n\\begin{eqnarray}\n \\langle e^{i\\Delta \\varphi_s }\\rangle \\to \\exp \\left(2ie V_s t\\over \\hbar \\right)\n=\\exp \\left(2ie I_s Rt\\over \\hbar \\right)=\\exp \\left(ir {I_s t\\over e} \\right)~.\n          \\label{asyTr}    \\end{eqnarray} \nThis asymptotic behavior really takes place for a strictly periodic sequence of current pulses with period\n$T_0=e\/|I_s|$. In this case the phase variation for the time interval $nT_0 <t\n< (n+1)T_0$ ($n$ is integer) is:\n\\begin{eqnarray}\n\\varphi(t)=r \\left[n +{1-e^{-(t-nT_0)\/\\tau} \\over1-e^{-T_0\/\\tau}}  \\right]  ~.           \n\\end{eqnarray}\nIn the limit $|I_s| \\to \\infty$ ($T_0 \\to 0$) the maximum deviation of the phase from the linear function\n$\\varphi(t)=rt \/T_0$ is of the order $r T_0^2\/\\tau ^2$, i.e., vanishes in this limit. This yields Eq. (\\ref{asyTr}).\n\nHowever, the expression (\\ref{Pois}) derived from the Poissonian statistics has another asymptotic\nbehavior for $t \\to \\infty$:\n\\begin{eqnarray}\n \\langle e^{i\\Delta \\varphi_s }\\rangle-1\n\\to  \\exp\\left[ \\left(e^{ir}-1\\right){I_s t\\over e}\\right] ~.\n        \\label{asyPoi}       \\end{eqnarray}\nThus the $IV$ curve of the\nJosephson junction essentially depends on whether the current pulses through the parallel (noise) junction are strictly\nperiodical in time, or completely random.  This proves strong sensitivity  to counting statistics.\n\nNote that the right-hand side of Eq. (\\ref{asyPoi}) coincides with the characteristic function\n$\\chi(r)=\\sum_n P_ne^{ir n}$ for the Poissonian distribution $P_n=e^{-\\bar{n}}\\sum_n \\bar{n}^n\/n!$ with\n$\\bar{n}=I_st\/e$. This is a natural result since  the number of phase jumps $n=\\Delta \\varphi_s\/2\\pi$ during a\nlong time $t \\gg \\tau$ coincides with the number of electrons transmitted through the noise junction during\nthe same time interval. Equation (\\ref{asyPoi})  leads to a conclusion that in the case of the Poissonian\nstatistics shot noise affects the $IV$ curve even at high noise currents when one might expect suppression of\nnoise effects. This suppression takes place for usual methods of noise detection probing voltage fluctuations.\nIndeed calculating the voltage variance $\\langle\\Delta V^2 \\rangle$ for the Poissonian statistics,\n\\begin{eqnarray}\n \\langle \\Delta V^2 \\rangle=  \\langle  V^2 \\rangle- \\langle  V \\rangle^2\n= {e\\over 2 C}\\langle  V \\rangle~,\n              \\end{eqnarray} \n one obtains that the relative mean-square-root voltage fluctuation $ \\sqrt{\\langle \\Delta V^2 \\rangle\n}\/\\langle  V \\rangle$ vanishes at $V_s =\\langle  V \\rangle=I_s R \\to \\infty$. However, the\nCoulomb blockaded junction probes not voltage but phase, which is much more sensitive to\nfluctuations. This ``supersensitivity'' of the Coulomb blockaded junction to fluctuations explains why the\neffects of higher moments (or cummulants) are much stronger than in usual methods of noise detection\n\\cite{BKN,Reul,Rez}.\n\n\\section{Josephson junction, weak coupling limit, phase noise in the AC input} \\label{sec3}\n\nNow we want to demonstrate that the Coulomb blockaded junction can effectively probe not only shot noise\nbut other types of noise as well. As an example, phase fluctuations in a monochromatic\nelectromagnetic wave will be considered. This type of noise is very important for lasers \\cite{L}, but the\npresent analysis addresses  microwaves. Suppose that some microwave source provides an input signal described by phase variation at\nthe Josephson junction:\n\\begin{eqnarray}\n \\varphi_i=\\varphi_0 \\sin \\psi(t)=\\varphi_0 \\sin \\left[\\omega t+ \\sum_i \\gamma_i\n\\Theta(t-t_i)\\right]~,\n        \\label{phN}      \\end{eqnarray}\nwhere $\\gamma_i$ are random phase jumps at random moments of time $t_i$. Frequency of phase jumps is characterized by the\naverage time $\\tau_c$ between the jumps. In fact $\\tau_c$ is nothing else as the decoherence time of the microwave signal. We\nshall see that the $IV$ curve of the Josephson junction is very sensitive to the decoherence process. But in order to demonstrate\nit one should first consider the response of the Josephson junction to  a strictly monochromatic noiseless microwave\ninput.\n\n\\subsection{Response to a monochromatic AC input} \\label{AClin}\n\nFor a strictly monochromatic signal the phase  difference between the moments $t_0$ and $t_0-t$ is \n\\begin{eqnarray}\n\\Delta \\varphi_i(t_0,t)=\\varphi_i(t_0)-\\varphi_i(t_0-t)=\\varphi_0 [\\sin \\omega t_0-\\sin \\omega (t_0-t)]\n=\\varphi_0 [\\sin \\omega t_0(1-\\cos \\omega t)+\\cos \\omega t_0\\sin \\omega t]~.\n              \\end{eqnarray}\nIn full analogy with the previous cases, the current is given by\n\\begin{eqnarray}\nI=-{2 eE_J^2 \\over \\hbar^2}\\mbox{Im}\\left\\{\\int_0^\\infty dt e^{J_0(t)}\n\\sin \\Delta\n\\varphi_i(t_0,t)\\right\\}~, \n       \\end{eqnarray}\nbut now averaging over the moment $t_0$ is not done. The linearization with respect to the input gives\n\\begin{eqnarray}\nI\\approx -{2 eE_J^2 \\over \\hbar^2}\\mbox{Im}\\left\\{\\int_0^\\infty dt e^{J_0(t)}\n\\Delta\n\\varphi_i(t_0,t)\\right\\} \n= -\\varphi_0{2 eE_J^2 \\over \\hbar^2}\\mbox{Im}\\left\\{\\int_0^\\infty dt e^{J_0(t)}\n[\\cos \\omega t_0\\sin \\omega t +\\sin \\omega t_0(1-\\cos \\omega t)]\\right\\} \n \\label{curAC} \\nonumber \\\\\n= I_1 \\cos \\omega t_0 +I_2 \\sin \\omega t_0 ~.\n   \\label{curPh}   \\end{eqnarray}\nThus the tunneling current of Cooper pairs consists of two parts: one is out of phase and another is in phase with the\ninput voltage $ V_i=V_0 \\cos \\omega t$ (the reactive and the active responses)  where $V_0=\\hbar \\omega \\varphi_0\/2e $. Comparing\nthe active response,\n\\begin{eqnarray}\nI_1= -\\varphi_0{2 eE_J^2 \\over \\hbar^2}\\mbox{Im}\\left\\{\\int_0^\\infty dt e^{J_0(t)}\n\\sin \\omega t\\right\\} ~,\n              \\end{eqnarray}\nwith Eq. (\\ref{WC}) one can see that the ohmic current as a function of frequency can be directly received from the \n$IV$ curve $I(V)$  replacing voltage $V$ by $\\hbar \\omega\/2e$. Therefore the  active (ohmic)\nresponse is given by\n\\begin{eqnarray}\nI_1= \\varphi_0 I\\left(\\hbar \\omega \\over 2e\\right)\n= V_0 {2e  \\over \\hbar \\omega} I\\left(\\hbar \\omega \\over 2e\\right)~.\n              \\end{eqnarray}\nThus the ohmic AC linear conductance of the junction is a nonanalytic function of frequency proportional to $\n\\omega^{2\\rho -2}$. \n\nThe reactive response for frequencies $\\omega \\ll \\rho\/\\tau=1\/R_QC$ is \n\\begin{eqnarray}\nI_2=-\\varphi_0{2 eE_J^2 \\over \\hbar^2}\\mbox{Im}\\left\\{\\int_0^\\infty dt e^{J_0(t)}\n(1-\\cos \\omega t)\\right\\}\\approx -\\varphi_0{ eE_J^2 \\over\n\\hbar^2}\\mbox{Im}\\left\\{\\int_0^\\infty dt e^{J_0(t)} \\omega^2 t^2\\right\\}~ .\n      \\end{eqnarray}\nAfter rotation in the complex plane ($t \\to-iy$):\n\\begin{eqnarray}\nI_2=-\\varphi_0{eE_J^2 \\over \\hbar^2}\\omega^2\\int_0^\\infty y^2 dy e^{-\\pi \\rho y\/\\tau}\n=-\\varphi_0{2 eE_J^2 \\over \\hbar^2}\\omega^2\\left(\\tau\\over \\pi \\rho\\right)^3\n=-V_0{2e\\over \\hbar \\omega}{2 eE_J^2 \\over \\hbar^2}\\omega^2\\left(R_Q C\\over \\pi\n\\right)^3=-V_0\\omega C{E_J^2 C^2\\over 2e^4}~.\n         \\end{eqnarray}\nThe linear dependence on frequency points out that this current corresponds to the capacitance element of the\nelectrical circuit. Indeed, the effective circuit for the Josephson junction has the capacitance $C$ in parallel to the\ntunneling conductance element. But in the presence of the Josephson coupling the geometric capacitance should be\nreplaced by the effective capacitance (like the mass of an electron in a periodic potential should be replaced by the\neffective mass, see, e.g., Ref. \\onlinecite{SZ}). The current component $I_2$ is related with the difference between the\neffective and the geometric capacitance in the weak coupling limit. In order to check it one should calculate the\nsecond-order (with respect to the Josephson coupling) correction to the Coulomb energy $Q^2\/2C$:\n\\begin{eqnarray}\nE={Q^2\\over 2 C}+ \\sum _{Q_G\\neq 0}{V(Q_G)^2 \\over E_0(Q)-E_0(Q+Q_G)}~,\n         \\end{eqnarray}\nwhere summation is over the inverse-lattice ``wave-numbers'' (charges) $Q_G=2e n$ ($n$ is any integer).  For weak\nJosephson coupling only two terms  $ Q_G=\\pm 2e$ of the sum are important, and $V(\\pm 2e)=E_J\/2$. Then \n\\begin{eqnarray}\nE={Q^2\\over 2 C}+ {E_J^2\\over 4}\\left[{2 C \\over Q^2-(Q+2e)^2}+ {2C \\over\nQ^2-(Q-2e)^2}\\right]\n={Q^2\\over 2 C}+{E_J^2\\over 4}{e^2 C\\over e^4 -e^2Q^2}\\approx {Q^2\\over 2 C}+{E_J^2 C\\over 4e^2}-{E_J^2 C\\over\n4e^4}Q^2~.\n         \\end{eqnarray}\nThis yields for the effective capacitance:\n\\begin{eqnarray}\n{1\\over C^*}= {d^2 E\\over dQ^2}={1\\over C}-{E_J^2 C\\over 2e^4}~,\n        \\end{eqnarray}\nand it is evident that the reactive component $I_2$ of the tunneling current is determined by $\\Delta C=C^*-C$. It is\nuseful to rewrite the expression for this correction using  the\nAmbegoakar-Baratoff relation:\n\\begin{eqnarray}\n E_J={\\pi \\hbar \\Delta \\over 4e^2 R_T}={\\Delta_0 \\over 2}{R_Q\\over R_T}~,\n        \\end{eqnarray}\nwhere $\\Delta_0$ is the bulk superconducting gap. Then the capacitance correction is\n\\begin{eqnarray}\n\\Delta C=C {E_J^2 C^2 \\over 2e^4}=C{R_Q^2\\over R_T^2}{\\Delta_0^2 \\over 8e^4\/C^2}~.\n  \\label{Ccor}       \\end{eqnarray}\n\nNow we calculate the nonlinear (quadratic) effect of the AC input on the DC conductance. This requires averaging (over\nthe period of the AC input) of the quadratic term in the expression for the conductance:\n\\begin{eqnarray}\n\\Delta G=-{4 e^2E_J^2 \\over \\hbar^3}\\mbox{Im}\\left\\{\\int_0^\\infty t\\,dt e^{J_0(t)}\n\\langle \\cos \\Delta\\varphi_i \\rangle\\right\\} \n\\approx {2 e^2E_J^2 \\over \\hbar^3}\\mbox{Im}\\left\\{\\int_0^\\infty t\\,dt\ne^{J_0(t)}\\langle  \\Delta \\varphi_i^2 \\rangle\\right\\} \\nonumber \\\\\n=\\varphi_0^2{2 e^2E_J^2 \\over \\hbar^3}\\mbox{Im}\\left\\{\\int_0^\\infty t\\,dt\ne^{J_0(t)}(1-\\cos \\omega t)\\right\\}~.\n   \\label{DG}   \\end{eqnarray}\nThe expansion in $\\omega$ contains only the terms even in $t$. The integrals over these terms vanish after rotation in\nthe plane of complex time. This means that $\\Delta G$ is determined by a very small nonanalytic contribution from long\ntimes \\cite{FBS}. The presence of\nnoise essentially changes the situation as we shall see in the next subsection.\n\n\\subsection{Phase noise}\n\nIn the presence of the phase noise, when the phase varies according to Eq. (\\ref{phN}), the squared difference between the phases\nat the moments of time $t_0$ and\n$t_0-t$ averaged over the time $t_0$ is given by\n\\begin{eqnarray}\n\\langle \\Delta \\varphi_i(t_0,t)^2\\rangle =\\varphi_0^2 \\langle[\\sin \\psi(t_0)-\\sin \\psi\n(t_0-t)]\\rangle^2 =\\varphi_0^2 \\{1-\\langle\\cos[\\psi(t_0)-\\psi (t_0-t)]\\rangle\\}\\, .\n    \\end{eqnarray}\nThe average $\\langle\\cos[\\psi(t_0)-\\psi (t_0-t)]\\rangle$ does not vanish only if there is no random phase jumps in the time\ninterval $t$. Assuming the  Poisson distribution the probability of the absence of random phase jumps in the interval $t$\nis\n$e^{-t\/\\tau_c}$, where\n$\\tau_c$ is the average period between random phase jumps. Then one obtains that\n\\begin{eqnarray}\n\\langle \\Delta \\varphi_i(t_0,t)^2\\rangle  =\\varphi_0^2(1- \\cos \\omega t e^{-t\/\\tau_c})~.\n    \\end{eqnarray}\nFinally the correction to the DC conductance [compare with Eq. (\\ref{DG})] is:\n\\begin{eqnarray}\n\\Delta G=\\varphi_0^2{2 e^2E_J^2 \\over \\hbar^3}\\mbox{Im}\\left\\{\\int_0^\\infty t\\,dt\ne^{J_0(t)}(1-\\cos \\omega t e^{-t\/\\tau_c})\\right\\}\n\\approx \\varphi_0^2{2 e^2E_J^2 \\over \\hbar^3 \\tau_c}\\mbox{Im}\\left\\{\\int_0^\\infty\nt^2\\,dt e^{J_0(t)}\\right\\}~.\n       \\end{eqnarray}\nAfter rotation in the complex plane the imaginary part of the integral does not vanish, and\n\\begin{eqnarray}\n\\Delta G=\\varphi_0^2{2 e^2E_J^2 \\over \\hbar^3 \\tau_c}\\int_0^\\infty\ny^2\\,dy e^{-\\pi \\rho y\/\\tau}=\\varphi_0^2{4 e^2E_J^2 \\tau^3\\over\\pi^3\n\\rho^3 \\hbar^3 \\tau_c}=\\varphi_0^2{E_J^2 C^3\\over 2 e^4 \\tau_c}\n=V_0^2{2E_J^2 C^3\\over  e^2 \\hbar ^2 \\omega^2 \\tau_c}~.\n      \\end{eqnarray}\nThis contribution to the zero-bias conductance essentially exceeds the small nonanalytic contribution from the\nstrictly monochromatic microwave and provides direct information on the decoherence time $\\tau_c$. \n\n\\section{Normal junction, shot noise}  \\label{sec4}\n\n\n\\subsection{The $IV$ curve in the time-domain formulation (without shot noise)}\n\nThis subsection reminds the $P(E)$ theory for a normal Coulomb blockaded junction \\cite{IN}. The current is given\nby \n\\begin{equation}\n  I= { e}[\\Gamma^+(V)-\\Gamma^-(V)] ~,\n        \\end{equation}\nwhere the forward tunneling probability is\n\\begin{equation}\n\\Gamma^+(V) = {1\\over e^2 R_T}\\int_{-\\infty}^\\infty dE \\frac{E}{1\n-\\exp\\left(-\\frac{E}{T}\\right)}P(eV-E)~.\n    \\label{ProbE} \\end{equation}\nHere $P(E)$ is given by Eq. (\\ref{P-E}). The backward tunneling rate is $\\Gamma^- (V)=\\Gamma^+(-V)$.\nSince now we consider tunneling of single electrons (instead of Cooper pairs) the correlation function is given by\n\\begin{equation}\nJ_0(t) =\\langle[ \\varphi(t) -\\varphi(0)]\\varphi(0)\\rangle\n= 2\\int_{-\\infty}^\\infty {d\\omega \\over \\omega} \\frac{\\mbox{Re} \nZ(\\omega)}{R_K} \\frac{e^{-i\\omega t} -1}{1-e^{-\\beta \\hbar \\omega}}~,\n      \\end{equation}\nwhere the single electron quantum resistance $R_K=h\/e^2$ replaces the Cooper-pair quantum resistance $R_Q=h\/2e^2$ in\nEq.~(\\ref{phaseCor}). In the case of purely ohmic environment, $Z(\\omega)=R\/(1+i\\tau \\omega)$, \none can use the time-domain formulation \\cite{JED} fulfilling integration over energy $E$ in Eq. (\\ref{ProbE}):\n\\begin{equation}\n\\Gamma^+(V) = {1\\over \\pi \\hbar e^2 R_T}\\mbox{Re}\\left\\{\\int_0^\\infty dt\n\\gamma(t)\\exp\\left[ J_0(t) + \\frac{ieVt}{\\hbar}\\right] \\right\\}  ~.\n    \\label{ProbT}\\end{equation}\nHere $\\tau=RC$ as before, and\n\\begin{equation}\n\\gamma(t) = \\int_{-\\infty}^\\infty dE \n\\frac{E}{1-\\exp\\left(-\\frac{E}{T}\\right)}\n\\exp\\left( - \\frac{iEt}{\\hbar}\\right) =i\\pi\\hbar ^2 {d\\over dt} \\delta(t)\n- {\\pi^2 \\over \\beta^2} \\frac{1}{\\sinh ^2 (\\pi t\/ \\hbar \\beta)}~.\n   \\end{equation}\nThen after some simple algebra the current is \n\\begin{eqnarray}\n  I= { e}[\\Gamma^+(V)-\\Gamma^-(V)]= {1\\over R_T} \\left[\nV  + {2\\pi \\over e\\hbar \\beta^2} \\mbox{Im}\\left\\{  \\int_0^\\infty \n\\frac{dt}{\\sinh ^2 (\\pi t\/ \\hbar \\beta)} e^{J_0(t)}\\sin \\frac{eVt}{\\hbar}\\right\\} \\right] ~.\n        \\end{eqnarray}\nAt $T=0$:\n\\begin{eqnarray}\n  I= {1\\over R_T} \\left[V  + {2 \\hbar \\over \\pi e } \\mbox{Im}\\left\\{ \\int_0^\\infty \n\\frac{dt}{t ^2 } e^{J_0(t)}\\sin \\frac{eVt}{\\hbar} \\right\\}\\right] ~,\n    \\label{i-v}    \\end{eqnarray}\nand the differential conductance is\n\\begin{equation}\n G= {dI \\over dV}=  {1\\over R_T} \\left[\n1 + {2\\over \\pi } \\mbox{Im}\\left\\{ \\int_0^\\infty \n\\frac{dt}{t} e^{J_0(t)}\\cos \\frac{eVt}{\\hbar}\\right\\} \\right] ~.\n        \\end{equation}\nFor $\\rho=R\/R_K \\gg 1$ in the limit $V \\to 0$ the conductance vanishes:\n\\begin{equation}\n G_0= {1\\over R_T} \\left[1  - {2\\over \\pi }  \\int_0^\\infty \\frac{dt}{ t}\\sin \\left( \\pi \\rho {t\\over\n\\tau}\\right)\n \\right] \\to  0~,\n          \\end{equation}\nbut there remains a small nonanalytic contribution, which originates from long times: $G_0 \\propto V^{2\\rho}$.\n\n\\subsection{Effect of shot noise}\n\nRepeating derivation of the  effect of the random phase from shot noise, which was done above for the Josephson junction,\none receives that the effect on the normal tunnel junction is similar: the  linear phase from the constant voltage,\n$eVt\/\\hbar$, must be replaced  by the phase $eVt\/\\hbar + \\Delta \\varphi_s$, where $\\Delta \\varphi_s=\\varphi_s(t_0)\n-\\varphi_s(t_0-t)$ is the phase difference produced by the current through the noise junction. Note that the Josephson\nrelation between the voltage and the phase from shot noise is different from that for the Josephson junction by the factor\nof 2:\n$V_s(t)=(\\hbar\/e) d\\varphi_s(t)\/dt_s$.  Then the probability of the forward tunneling is [instead of Eq. (\\ref{ProbT})]\n\\begin{equation}\n\\Gamma^+(V) = {1\\over \\pi \\hbar e^2 R_T}\\mbox{Re}\\left\\{\\int_0^\\infty dt\n\\gamma(t)\\left\\langle\\exp\\left( J(t) + \\frac{ieVt}{\\hbar}+i\\Delta \\varphi_s\\right) \\right\\rangle \\right\\} ~,\n    \\end{equation}\nand the total current is \n\\begin{eqnarray}\n I= { e}[\\Gamma^+(V)-\\Gamma^-(V)]= {1\\over R_T} \\left[\nV + V_s  + {2\\pi \\over e\\hbar \\beta^2}\\mbox{Im}\\left\\{  \\int_{0}^\\infty \n\\frac{dt}{\\sinh ^2 (\\pi t\/ \\hbar \\beta)} e^{J(t)}\\left\\langle\\sin\n\\left( \n\\frac{eVt}{\\hbar}+\\Delta \\varphi_s\\right) \\right\\rangle \\right\\}\\right]~.\n        \\end{eqnarray}\nHere $ V_s =I_sR$ is the average voltage generated by the current through the noise junction.  In\nthe\n$T=0$ limit the current is:\n\\begin{eqnarray}\n I= {1\\over R_T} \\left[\nV +V_s + {2\\hbar  \\over \\pi e} \\mbox{Im}\\left\\{ \\int_{0}^\\infty \n\\frac{dt}{t^2} e^{J(t)}\\left\\langle\\sin\n\\left( \n\\frac{eVt}{\\hbar}+\\Delta \\varphi_s\\right) \\right\\rangle \\right\\}\\right]~.\n        \\end{eqnarray}\n\nFor the normal junction one should replace $r$ by $2r$ in Eqs. (\\ref{sin}) and (\\ref{cos}) if the single-electron\nquantum resistance $R_K$ is used in $r=\\pi \\rho=\\pi R\/R_K$. We need the approximate expressions for the phase\ncorrelator at\n$t\\ll\n\\tau$:\n\\begin{eqnarray}\n\\langle \\sin \\Delta \\varphi_s \\rangle \\approx {I_s \\tau\\over e}\\left(\\mbox{si} {2rt\\over \\tau }  +{\\pi \\over 2} \n\\right)  ={I_s \\tau\\over e}\\mbox{Si} {2rt\\over \\tau }  ~,            \n\\end{eqnarray}\n\\begin{eqnarray}\n(\\langle \\cos \\Delta \\varphi_s \\rangle-1)\\approx {I_s \\tau\\over e}\\left(\\mbox{ci}\n{2rt\\over \\tau } -\\gamma-\\ln {2rt\\over \\tau }\\right)~.\n              \\end{eqnarray}\nWith help  of these expressions one receives the expressions for the ratchet  current,\n\\begin{eqnarray}\n I_0= {1\\over R_T} \\left[V_s + {2\\hbar  \\over \\pi e}  \\int_{0}^\\infty \n\\frac{dt}{t^2} e^{J_R(t)}\\sin J_I(t)\\left\\langle \\sin\n\\Delta \\varphi(t_s,t)\\right\\rangle \\right] \\approx\n {1\\over R_T} \\left[V_s - {2\\hbar  \\over \\pi e} {I_s \\tau\\over e}\n\\int_{0}^\\infty \n\\frac{dt}{t^2}  \\sin {rt\\over \\tau }\n\\mbox{Si} {2 rt\\over \\tau }\\right] \\nonumber \\\\\n={1\\over R_T} \\left[V_s - (1+\\ln 2)I_s {\\hbar  \\over e^2} \nr\\right]=0.153 {V_s\\over  R_T}~,\n        \\end{eqnarray}\nand for the shot-noise contribution to the conductance:\n\\begin{eqnarray}\nG_s={d I\\over dV}= {1\\over R_T} \\left[1 + {2 \\over \\pi }  \\int_{0}^\\infty \n\\frac{dt}{t} e^{J_R(t)}\\sin J_I(t)\\left\\langle\\cos\n\\Delta \\varphi(t_s,t)\\right\\rangle \\right] \\nonumber \\\\\n\\approx\n-{1\\over R_T}  {2 \\over \\pi } {|I_s| \\tau\\over e} \\int_{0}^\\infty \n\\frac{dt}{t}\\sin {rt\\over \\tau }\\left(\\mbox{ci}\n{2rt\\over \\tau } -\\gamma-\\ln {2rt\\over \\tau }\n\\right)\n={0.693\\over\nR_T}   {|I_s|\\tau\\over e} ={0.693\\over\nR_T}   {|V_s|\\over e\/C}~.\n        \\end{eqnarray}\n\n\\subsection{Response of the normal junction to an AC input}\n\nThe normal Coulomb blockade junction is sensitive to phase noise in an AC input as well. First let us consider the\nlinear response to a monochromatic input using analogy with the case of the Josephson junction (Sec. \\ref{AClin}):\n\\begin{eqnarray}\n  I= {1\\over R_T} \\left[V_i  + {2 \\hbar \\over \\pi e }  \\int_0^\\infty \n\\frac{dt}{t ^2 } \\mbox{Im} \\{e^{J_0(t)}\\}\\sin\\Delta \\varphi_i(t_0,t) \\right]\n\\approx  {1\\over R_T} \\left[V_i  + {2 \\hbar \\over \\pi e }  \\int_0^\\infty \n\\frac{dt}{t ^2 } \\mbox{Im} \\{e^{J_0(t)}\\}\\Delta \\varphi_i(t_0,t) \\right]  \\nonumber \\\\\n={1\\over R_T} \\left[V_0 \\cos \\omega t_0  -\\varphi_0 {2 \\hbar \\over \\pi e }  \\int_0^\\infty \n\\frac{dt}{t ^2 } \\mbox{Im} \\{e^{J_0(t)}\\}[\\cos \\omega t_0\\sin \\omega t +\\sin \\omega t_0(1-\\cos \\omega\nt)] \\right]\n= I_1 \\cos \\omega t_0 +I_2 \\sin \\omega t_0 ~,\n     \\end{eqnarray}\nwhere $V_i=V_0 \\cos \\omega t_0$ with $V_0=\\hbar \\omega \\varphi_0 \/e$. The ohmic response is\n\\begin{eqnarray}\n I_1= {V_0\\over R_T}  + {2\\hbar  \\over \\pi e R_T}\\varphi_0  \\int_{0}^\\infty \n\\frac{dt}{t^2} \\mbox{Im} \\{e^{J_0(t)}\\}\\sin\\omega t\n= \\varphi_0  I\\left(\\hbar \\omega \\over e\\right)\n=V_0{e\\over \\hbar \\omega}I\\left(\\hbar \\omega \\over\ne\\right)~,\n        \\end{eqnarray}\nwhere the $IV$ curve $I(V)$ of the normal junction without AC input was used [Eq. (\\ref{i-v})] with $V$ replaced by $\\hbar\n\\omega\/e$.\n\nThere is also the reactive response:\n\\begin{eqnarray}\n I_2= {2\\hbar  \\over \\pi e R_T}\\varphi_0 \\mbox{Im}\\left\\{ \\int_{0}^\\infty \n\\frac{dt}{t^2} e^{J(t)}(1-\\cos\\omega t) \\right\\} \\approx\n{2\\hbar  \\over \\pi e R_T}\\varphi_0 \\omega^2 \\mbox{Im}\\left\\{\\int_{0}^\\infty \ndt  e^{J(t)}\\right\\} \\nonumber \\\\\n=-\\omega V_0{2  \\over \\pi R_T}  \\int_{0}^\\infty \ndy  e^{-\\pi \\rho t\/\\tau}=-\\omega V_0{2 \\tau \\over \\pi^2 \\rho R_T} \n=-\\omega C V_0{2 R_K \\over \\pi^2 R_T} ~.\n        \\end{eqnarray}\nThe reactive response is a result of the Coulomb blockade correction to the geometric capacitance:\n\\begin{eqnarray}\n \\Delta C= C {2 R_K \\over \\pi^2 R_T}\n        \\end{eqnarray}\nsimilar to the correction for the Josephson junction, Eq. (\\ref{Ccor}). \n\n\nQuadratic correction to the DC zero-bias conductance is also sensitive to the decoherence time $\\tau_c$:\n\\begin{eqnarray}\n\\Delta G\\approx -{1\\over R_T} {1 \\over \\pi } \\mbox{Im}\\left\\{ \\int_{0}^\\infty \n\\frac{dt}{t} e^{J_0(t)}\\left\\langle\\Delta \\varphi_i^2\\right\\rangle \\right\\}\n=-{1\\over R_T} {\\varphi_0^2 \\over \\pi } \\mbox{Im}\\left\\{ \\int_{0}^\\infty \n\\frac{dt}{t} e^{J_0(t)}(1-\\cos \\omega t e^{-t\/\\tau_c})\\right\\} \\nonumber \\\\\n\\approx -{1\\over R_T} {\\varphi_0^2 \\over \\pi \\tau_c} \\mbox{Im}\\left\\{ \\int_{0}^\\infty \ndt e^{J_0(t)}\\right\\} = {1\\over R_T} {\\varphi_0^2 \\over \\pi \\tau_c} \\int_{0}^\\infty \ndy e^{-\\pi \\rho\/\\tau}= {1\\over R_T} {\\varphi_0^2 \\tau\\over \\pi^2 \\rho \\tau_c} \n = {2\\over R_T} {V_0^2 C\\over \\pi\\hbar \\omega^2 \\tau_c}~.\n        \\end{eqnarray}\n\n\\section{Josephson junction, strong coupling limit, shot noise} \\label{sec5}\n\n\\subsection{Without shot noise}\n\n\\begin{figure\n  \\begin{center}\n    \\leavevmode\n    \\includegraphics[width=0.8\\linewidth]{Duality.eps}\n    \\bigskip\n    \\caption{Strong coupling limit and environment duality. The dashed squares \nshow the circuit elements, which present the Josephson and the noise junctions with  the capacitance $C=C_J+C_s$. The\ncurrent source $I_\\Sigma=I+I_s$ provides the constant current $I$ and the noisy current $I_s$ in parallel. a)\nThe dual weak-coupling case.  The inductance\n$L_J$ of the Josephson junction is negligible in the weak-coupling limit. b) The dual strong-coupling limit. c) The\nstrong-coupling case taking into account the capacitance\n$C_0$ and the resistance $R_0$ of the leads.}\n  \\label{fig3}\n  \\end{center}\n  \\end{figure}\n\n In the case of equilibrium noise the strong coupling limit can be investigated using the\nduality relations, which connect the parameters of the Josephson junction in the weak coupling and the strong\ncoupling limits \\cite{SZ,Weiss}:\n\\begin{equation}\nE_J \\rightarrow 2 \\Delta~,~~V \\rightarrow {h \\over 4e^2}I = R_Q I~,~~\\varphi\n\\rightarrow {\\pi Q \\over e}~,~~~\\rho={R\\over R_Q} \\rightarrow \\tilde\n\\rho={R_Q\\over R}~,~~~Z \\rightarrow Y~.\n      \\label{du} \\end{equation}\nHere $Y(\\omega)$ is the admittance of the electric circuit, and $\\Delta$ is the matrix element for a transition between adjacent\nstates of the Josephson junction with localized phases $2\\pi n$ and $2\\pi (n+1)$ ($\\hbar\/\\Delta$ is the life time at the state,\nwhich corresponds to some localized phase). Using these relations the\n$IV$ curve in the weak coupling limit, Eq. (\\ref{WC}), transforms to the dual $VI$ curve in the strong-coupling\nlimit:\n\\begin{eqnarray}\nV=-{4\\pi \\Delta^2 \\over e \\hbar}\\int_0^\\infty dt \\mbox{Im} \\left\\{e^{\\tilde J_0(t)}\n\\right\\} \\sin \\left(2eR_Q It\\over \\hbar\\right)\n=-{4\\pi \\Delta^2 \\over e \\hbar}\\int_0^\\infty dt \\mbox{Im} \\left\\{e^{\\tilde J_0(t)}\n\\right\\} \\sin \\left(\\pi It\\over e\\right)~.\n \\end{eqnarray}\nThe charge-charge correlator [see Eq. (149) in Ref. \\onlinecite{IN}] is given by the relation dual to Eq.\n(\\ref{phaseCor}):\n\\begin{eqnarray}\n\\tilde J_0(t)=2R_Q  \\int_0^\\infty\n{d\\omega \\over \\omega} {\\mbox{Re}  Y(\\omega)}\n \\left[\\coth\\left({1\\over 2}\\beta \\hbar \\omega\\right)(\\cos \\omega t -1)\n-i\\sin \\omega t\\right] ~.\n    \\label{Q-IN}  \\end{eqnarray}\n \nIn order to exploit duality completely, i.e., not only for the general expression for the $IV$ curve but also for the\nspecific results for ohmic environment with the effective impedance $Z=(1\/R+i\\omega C )^{-1}$, one should find a\ncorresponding ``dual'' environment for the strong-coupling case. In\nthe weak-coupling case the finite relaxation time $\\tau=RC$ is provided by\nthe imaginary element of the impedance $Z$, which is the capacitance $C=C_J+C_s$ of the two junctions (without the\nnoise source $C_s=0$). In the strong coupling case the both imaginary elements of the Josephson junction (capacitance\nand inductance) do not contribute to the admittance (see Fig. \\ref{fig3}b). In order to get a finite relaxation time\ndual to the time $\\tau=RC$ of the weak coupling limit one should add to the effective electric circuit\nthe inductance $L$ in series with the ohmic shunt resistance $R$ (Fig. \\ref{fig3}b). Then the admittance\ndual to the impedance $Z$ of the weak coupling limit is \n\\begin{equation}\n Y(\\omega)=\\frac{1}{R +i\\omega L}=\\frac{1\/R}{1+i\\omega \\tau_L}~,\n      \\end{equation}\nwhere the circuit relaxation time $\\tau_L =L\/R$ is dual to the relaxation time $\\tau=RC$ in the weak\ncoupling limit. Thus duality relations Eq. (\\ref{du}) should by supplemented by the ``environment\nduality'' relation $C \\rightarrow L$. Note, however, that introduction of finite inductance $L$ is not\nthe only way to achieve a finite relaxation time. For example, Sch\\\"on and Zaikin \\cite{SZ} suggested\nthat the period of Josephson plasma oscillation $\\sim \\hbar\/\\sqrt{E_J E_c}$ should play the role of\nrelaxation time dual to the time $RC$ in the weak coupling limit. \n\n\\subsection{With shot noise}\n\nThe duality concept is valid only for equilibrium noise when symmetry\nbetween current and voltage or impedance and admittance takes place. In the case of shot noise this\nsymmetry is broken since shot noise is a {\\em current} noise, which produces asymmetric effects on the\ncurrent and on the voltage. Though we can still use the duality for description of the equilibrium\ncontribution to the noise (this contribution cannot be ignored even if our goal is the effect of shot\nnoise), the effect of shot noise on the $IV$ curve is essentially different in the weak and the strong\ncoupling limits.\n\nIn the strong-coupling limit the effect of environment is governed by charge fluctuations. Whereas the phase\nfluctuations were determined by the voltage fluctuations via the Josephson relation $\\hbar d\\varphi\/dt=2eV$, the charge\nfluctuations are determined by the current fluctuations via the relation\n$dQ\/dt =I$. Important is that the current $I$ should be the {\\em total current} of the part of the circuit, which describes the\nJosephson junction in the RCSJ model (inductance + capacitance + shunt resistance). Single-electron tunneling through the noise\njunction produces a pulse  of the total current. If the tunneling time is very short, the pulse can be approximated by the\n$\\delta$-function current peak. The effective circuit in  Fig. \\ref{fig3}b cannot broaden the original $\\delta$-function\ncurrent peak. This is in contrast to the voltage pulses, which were relevant for phase fluctuations in the weak coupling\nlimit: the $\\delta$-function current peak produced the voltage pulse of width $\\tau=RC$.\nMeanwhile, the finite width of the current pulse is of principal importance for the shot-noise effect  and therefore cannot be\nignored. Broadening of the current pulse can be determined by additional lump elements of the electric circuit, which are\nconnected with the ohmic resistance $R_0$ and the capacitance $C_0$ of wires connecting the current-noise source and the probe\nJosephson junction (see Fig. \\ref{fig3}c). Bearing in mind this electric circuit the single-electron tunneling at $t-0$ produces\nthe current pulse\n\\begin{equation}\n\\delta I (t)=\\mbox{sign}(I_s){e\\over \\tau_s} e^{-t\/\\tau_s}~,\n      \\end{equation}\nwhich brings the charge\n\\begin{equation}\n\\delta Q_s(t)=\\mbox{sign}(I_s)e(1- e^{-t\/\\tau_s})~.\n      \\end{equation}\nHere $\\tau_s=R_0C_0$. \nThe $VI$ curve with shot noise is given by\n\\begin{eqnarray}\nV=-{4\\pi \\Delta^2 \\over e \\hbar}\\int_0^\\infty dt \\mbox{Im} \\left\\{e^{\\tilde J_0(t)}\n\\right\\} \\left\\langle\\sin \\left({\\pi It\\over e}+{\\pi\\Delta Q_s\\over e}\\right)\\right\\rangle~,\n \\end{eqnarray}\nwhere $\\Delta Q_s=Q_s(t_0)-Q_s(t_0-t)$ and\n\\begin{eqnarray}\nQ_s(t)  =\\mbox{sign}(I_s)\\pi \\sum_i \\Theta (t-t_i)\\left [ 1-\ne^{-(t-t_i)\/\\tau}\\right]  ~.\n     \\end{eqnarray}\nThe shot-noise charge-charge correlators are found by averaging over the the time $t_0$. \n The expressions for $\\langle\\cos (\\pi \\Delta Q_s\/e)\\rangle$ and $\\langle\\sin\n(\\pi \\Delta Q_s\/e)\\rangle$ follow from those for $\\langle\\cos (\\Delta\n\\varphi_s)\\rangle$ and $\\langle\\sin (\\Delta \\varphi_s)\\rangle$ assuming $\\rho=1$ and using the duality\nrelations between the phase and the charge:\n\\begin{eqnarray}\n{e\\over I_s \\tau_s}\\left\\langle \\sin {\\pi\\Delta Q_s\\over e}\\right\\rangle = \n\\mbox{si} \\pi - \\mbox{si} \\left(\\pi\ne^{-t\/\\tau_s} \\right) +  \\mbox{si} \\left[\\pi\\left(1-\ne^{-t\/\\tau_s} \n\\right) \\right] +{\\pi \\over 2}~,\n       \\label{sinG}       \\end{eqnarray}\n\\begin{eqnarray}\n{e\\over |I_s |\\tau_s}\\left(\\left\\langle \\cos {\\pi\\Delta Q_s\\over e} \\right\\rangle-1\\right)= -\n\\mbox{ci} \\pi + \\mbox{ci} \\left(\\pi  e^{-t\/\\tau_s} \\right)-{t\\over\n\\tau_s} +\n\\mbox{ci} \\left[\\pi\\left(1- e^{-t\/\\tau_s} \n\\right) \\right] -\\gamma-\\ln \\left[\\pi\\left(1-\ne^{-t\/\\tau_s} \n\\right) \\right] ~.\n    \\label{cosS}           \\end{eqnarray}\nIn contrast to the weak-coupling case where the single relaxation time $\\tau$ characterized both the\nequilibrium and the shot-noise correlators, in the strong coupling case the equilibrium charge correlator is\ncharacterized by the time $\\tau_L$, whereas the shot-noise charge correlator has the other characteristic time\n$\\tau_s$. We expect that $\\tau_s \\ll \\tau_L$. Therefore we should consider the limit $t \\gg \\tau_s$ in Eqs.\n(\\ref{sinG}) and (\\ref{cosS}):\n\\begin{eqnarray}\n\\left\\langle \\sin {\\pi\\Delta Q_s\\over e}\\right\\rangle =(2 \\mbox{si} \\pi  +  \\pi) {I_s\\over e} \\tau_s\n =3.704 {I_s\\over e} \\tau_s~,\n              \\end{eqnarray}\n\\begin{eqnarray}\n\\left\\langle \\cos {\\pi\\Delta Q_s\\over e} \\right\\rangle-1= -2{|I_s|\\over e} t ~.\n         \\end{eqnarray}\n\nWe expand the shot noise contribution with respect to the current $I$:\n\\begin{eqnarray}\n\\Delta V=-{4\\pi \\Delta^2 \\over e \\hbar}\\int_0^\\infty dt \\mbox{Im} \\left\\{e^{\\tilde\nJ_0(t)}\n\\right\\}\\left[ \\sin \\left({\\pi It\\over e}\\right) \\left(\\left\\langle\\cos{\\pi\\Delta Q_s\\over e}\n\\right\\rangle-1\\right) +\\cos \\left({\\pi It\\over e}\\right) \\left\\langle\\sin{\\pi\\Delta Q_s\\over e}\n\\right\\rangle\\right]\\nonumber \\\\\n\\approx -{4\\pi \\Delta^2 \\over e \\hbar}\\int_0^\\infty dt \\mbox{Im} \\left\\{e^{\\tilde\nJ_0(t)}\n\\right\\}\\left[ {\\pi It\\over e} \\left(\\left\\langle\\cos{\\pi\\Delta Q_s\\over e}\n\\right\\rangle-1\\right) +\\left\\langle\\sin{\\pi\\Delta Q_s\\over e}\n\\right\\rangle\\right] = V_0+R_s I ~.\n \\end{eqnarray}\nRotating the integration path in the complex plane ($t\\to -iy$)  we obtain the\nratchet voltage $V_0$ and the zero-bias resistance $R_s $:\n\\begin{eqnarray}\nV_0 =-{4\\pi \\Delta^2 \\over e \\hbar}\\int_0^\\infty dt \\mbox{Im} \\left\\{e^{\\tilde\nJ_0(t)}\n\\right\\}\\left\\langle\\sin{\\pi\\Delta Q_s\\over e}\n\\right\\rangle\n\\nonumber \\\\\n=-{14.8\\pi \\Delta^2 \\over e^2 \\hbar}I_s \\tau_s\\int_0^\\infty dt \\mbox{Im}\n\\left\\{e^{\\tilde J_0(t)}\n\\right\\}={14.8\\pi \\Delta^2 \\over e^2 \\hbar}I_s \\tau_s\\int_0^\\infty dy\ne^{-\\pi\\tilde \\rho y\/\\tau_L}={14.8 \\Delta^2 \\over e^2\n\\hbar}{ \\tau_L\\over \\tilde \\rho }I_s \\tau_s={7.4 \\pi\\Delta^2 \\over e^4}{L\\tau_s\\over R_Q^2 }I_s ~,\n \\end{eqnarray}\n\\begin{eqnarray}\nR_s =-{4\\pi^2 \\Delta^2 \\over e^2 \\hbar}\\int_0^\\infty t\\, dt \\mbox{Im} \\left\\{e^{\\tilde\nJ_0(t)}\n\\right\\}\\left(\\left\\langle\\cos{\\pi\\Delta Q_s\\over e}\n\\right\\rangle-1\\right) \n\\nonumber \\\\\n=| I_s| {8\\pi^2 \\Delta^2 \\over e^3 \\hbar}\\int_0^\\infty t^2\\,dt \\mbox{Im}\n\\left\\{e^{\\tilde J_0(t)}\\right\\}=| I_s| {8\\pi^2 \\Delta^2 \\over e^3 \\hbar}\\int_0^\\infty\ny^2\\,dy e^{-\\pi\\tilde \\rho y\/\\tau_L} =| I_s| {16\\pi^2 \\Delta^2 \\over e^3 \\hbar}\\left(\\tau_L \\over \\pi\n\\tilde\n\\rho\\right)^3 =| I_s| {8 \\Delta^2 \\over  e^5 }{L^3 \\over R_Q^4}~. \n \\end{eqnarray}\nOne can see that the even effect (resistance $R_s$) remains finite in the limit $\\tau_s \\to 0$. In contrast, the\nratchet effect, which is an odd effect, vanishes. More generally, any odd effect should vanish in the limit $\\tau_s\n\\to 0$. This is because in this limit tunneling events lead to instantaneous transfers of charge $\\pm e$. The odd\neffect should depend on the sign of transferred charge, but the current cannot depend on whether the charge $+e$ or\n$-e$ suddenly appears at the Josephson junction due to periodicity of the charge correlator with the period\n$2e$: the instantaneous transfers of the charges $+e$ and $-e$ must produce identical effects. Only finite duration of\nthe current pulse (finite $\\tau_s$) can lead to an odd effect (ratchet voltage $V_0$ as an example).\n\nThis interesting feature of the strong-coupling case becomes even more pronounced if shot noise originates\nfrom the current of Cooper pairs. This can be realized if the noise junction is a SIN junction in the regime of\nthe Andreev reflection or a Josephson junction in the weak coupling limit in the regime of incoherent tunneling\nof Cooper pairs. In the both cases the tunneling event is accompanied by the charge jump\n$Q_s=2e(1-e^{-t\/\\tau_s})\\mbox{sign}(I_s)$ and, in analogy with Eqs. (\\ref{sinG}) and (\\ref{cosS}), the odd and\nthe even charge correlators can be obtained from the phase-phase correlators using duality and assuming\n$\\rho=2$:\n\\begin{eqnarray}\n{e\\over I_s \\tau_s}\\left\\langle \\sin {\\pi\\Delta Q_s\\over e}\\right\\rangle = \n-\\mbox{si}(2 \\pi) + \\mbox{si} \\left(2\\pi e^{-t\/\\tau_s} \\right) +  \\mbox{si} \\left[2\\pi\\left(1-\ne^{-t\/\\tau_s} \\right) \\right] +{\\pi \\over 2}~,\n              \\end{eqnarray}\n\\begin{eqnarray}\n{e\\over |I_s| \\tau_s}\\left(\\left\\langle \\cos {\\pi\\Delta Q_s\\over e} \\right\\rangle-1\\right)= \\mbox{ci}(2 \\pi) -\n\\mbox{ci} \\left(2\\pi  e^{-t\/\\tau_s} \\right)-{t\\over\n\\tau_s} +\\mbox{ci} \\left[2\\pi\\left(1- e^{-t\/\\tau_s} \\right) \\right] -\\gamma-\\ln \\left[2\\pi\\left(1-\ne^{-t\/\\tau_s} \\right) \\right] ~.\n               \\end{eqnarray}\nIn the limit $t\\gg \\tau_s$\n\\begin{eqnarray}\n{e\\over I_s \\tau_s}\\left\\langle \\sin {\\pi\\Delta Q_s\\over e}\\right\\rangle =2\\pi  e^{-t\/\\tau_s}~,\n              \\end{eqnarray}\n\\begin{eqnarray}\n{e\\over |I_s |\\tau_s}\\left(\\left\\langle \\cos {\\pi\\Delta Q_s\\over e}\\right \\rangle-1\\right) \n= 2[\\mbox{ci} (2\\pi) - \\gamma-\\ln\n(2\\pi)] +\\pi^2e^{-2t\/\\tau_s} ~.\n               \\end{eqnarray}\nCalculating the shot-noise contribution to\nthe voltage one cannot expand in $t\/\\tau_s$ but can expand in\n$t\/\\tau_L$ and in the current $I$:\n\\begin{eqnarray}\n\\Delta V =-{4\\pi \\Delta^2 \\over e \\hbar}\\int_0^\\infty dt \\mbox{Im} \\left\\{e^{\\tilde J_0(t)}\n\\right\\}\\left[ \\left(\\left \\langle \\cos{\\pi\\Delta Q_s\\over e}\\right\\rangle-1\\right))\\sin \\left({\\pi It\\over e}\\right) +\\left\n\\langle\n\\sin {\\pi\\Delta Q_s\\over e}\\right \\rangle\\cos \\left({\\pi It\\over e}\\right) \\right] \\nonumber \\\\\n\\approx {4\\pi^2 \\Delta^2 \\over e \\hbar}{\\tilde \\rho\\over \\tau_L}\\int_0^\\infty t\\,dt \\left[\\left (\\left \\langle\n\\cos\n{\\pi\\Delta Q_s\\over e}\\right \\rangle-1\\right){\\pi It\\over e} +\\left \\langle \\sin {\\pi\\Delta Q_s\\over e}\n\\right\\rangle\\right]=V_0+R_s I~.\n \\end{eqnarray}\nIntegrating by parts one obtains the ratchet voltage \n\\begin{eqnarray}\nV_0 = {4\\pi^2 \\Delta^2 \\over e \\hbar}{\\tilde \\rho\\over \\tau_L}\\int_0^\\infty t\\,dt \\left\\langle \\sin\n{\\pi\\Delta Q_s\\over e} \\right \\rangle={8\\pi^3 \\Delta^2 \\over e^2 \\hbar}{\\tilde \\rho\\over \\tau_L}I_s\n\\int_0^\\infty  e^{-t\/\\tau_s}{t^2\\over 2}\\,dt \\left\\{\\mbox{si}'(2\\pi e^{-t\/\\tau_s}) \\right. \\nonumber \\\\ \\left.\n-\\mbox{si}'\\left[2\\pi\\left(1- e^{-t\/\\tau_s} \\right) \\right]\\right\\} \n={2\\pi^2 \\Delta^2 \\over e^2 \\hbar}{\\tilde \\rho\\over \\tau_L}I_s\n\\tau_s^3\\int_0^\\infty x^2\\,dx \\left[\\sin(2\\pi e^{-x})-{\\sin\\left[2\\pi\\left(1- e^{-x} \\right) \\right]\\over\ne^x-1}\\right]={7.62\\pi^3 \\Delta^2  \\tau_s^3\\over e^4  L}  I_s ~,\n \\end{eqnarray}\nand the short noise contribution to the zero-bias resistance\n\\begin{eqnarray}\nR_s = {4\\pi^3 \\Delta^2 \\over e^2 \\hbar}{\\tilde \\rho\\over \\tau_L}\\int_0^\\infty t^2\\,dt  \\left(\\left\\langle\n\\cos{\\pi\\Delta Q_s\\over e} \\right\\rangle-1\\right)=- {4\\pi^3 \\Delta^2 \\over e^3 \\hbar}{\\tilde \\rho\\over\n\\tau_L}|I_s|\\int_0^\\infty {t^3\\over 3}\\,dt \\left[ \\mbox{ci}'(2\\pi e^{-t\/\\tau_s})2\\pi e^{-t\/\\tau_s}-1\n\\right.\\nonumber \\\\ \\left. +\\mbox{ci}'\\left[2\\pi\\left(1- e^{-t\/\\tau_s} \\right) \\right]2\\pi\ne^{-t\/\\tau_s} -{e^{-t\/\\tau_s} \\over 1-e^{-t\/\\tau_s}}\\right]\\nonumber \\\\\n= {4\\pi^3 \\Delta^2 \\tau_s^4\\over 3e^3 \\hbar}{\\tilde \\rho\\over \\tau_L}|I_s|\\int_0^\\infty x^3\\,dx\n\\left[1-\\cos(2\\pi e^{-x})  +{1-\\cos\\left[2\\pi\\left(1- e^{-x} \\right) \\right]\\over e^x-1}\\right] \n ={5.08\\pi^4\\Delta^2  \\tau_s^4\\over\ne^5L}|I_s| ~.\n \\end{eqnarray}\nThus shot noise from the current of Cooper pairs produces neither odd nor even effects\nif duration $\\tau_s$ of a single current pulse vanishes.\n\nWhat would happen if experimentalists were able to decrease the time $\\tau_s$ essentially? Eventually this\ntime would be comparable or even less that the intrinsic time, which determines duration of\nsingle-electron tunneling. Then the experimental observation of the zero-bias anomaly would provide an\ninformation on this intrinsic time. Up to now the tunneling time is mostly an issue of theoretical debates\n\\cite{trav,Hauge,Land}: Though some attempts to observe it were undertaken, but their interpretation is ambiguous (see\nRef. \\onlinecite{Land} and references therein). The subject is controversial and the consensus on how this time should\nbe defined is absent. Decreasing of $R_0$ and\n$C_0$ the experimentalists would shift this issue  from the pure theory to the measurement. How realistic could this\nperspective be? Presently it is difficult to achieve\n$R_0$ less than 100 $\\Omega$ and $C_0$ less then 1 fF. This yields $\\tau_s=R_0C_0 >10^{-13}$ s. On the other hand,\nwhatever disagreement on definition of the tunneling time, usually it is estimated as\n$10^{-15}~-~10^{-14}$ s. This yields the hope that further progress in miniaturization of the\nexperimental set up or searching for cases with longer tunneling time could open a new possibility to investigate\nthe tunneling time experimentally.\n\n\\section{Conclusions}\n\nWe have presented the theory of the  effect of shot noise from an independent source on the Coulomb blockaded\nJosephson junction in high-impedance environment. The analysis takes into account asymmetry of \nshot noise characterized by its odd moments. For high impedance environment the effect is so strong that the expansion in\nmoments is not valid and was not used in the analysis. Asymmetry of shot noise results in asymmetry of the $IV$\ncurve: the shift of the conductance minimum from the zero bias and the ratchet effect, which\nhave been observed experimentally \\cite{SN,exp}. At low noise currents (currents responsible for shot noise) the effect is\nproportional to the noise current independently of counting statistics of electron transport. However, at high noise\ncurrents the effect of noise on the $IV$ curve is very sensitive to electron counting statistics. This high\nsensitivity is explained by the fact that in contrast to usual methods of noise detection the Coulomb\nblockaded junction probes phase but not voltage.\n\nThe theory was generalized on  another type of noise (phase noise of a monochromatic AC input) and on a normal\nCoulomb blockaded tunnel junction, In the both cases the effect of noise strongly affects the Coulomb\nzero-bias anomaly of the $IV$ curve. The effect of shot noise on the superconducting Josephson junction in\nlow-impedance environment was also considered. If environment generates only the equilibrium Johnson-Nyquist\nnoise this case can be analyzed using the duality  relations with the case of the Coulomb blockaded Josephson\njunction in high-impedance environment. However, shot noise breaks duality relations between the two cases. An\ninteresting feature of the superconducting Josephson junction in low-impedance environment has been revealed:\nthe effect of shot noise on its $IV$ curve can give information on the time of electron tunneling through the\njunction responsible for shot noise.\n\nAltogether, the analysis demonstrates, that the  low-bias part of the $IV$ curves of tunnel junctions, both Coulomb\nblockaded and superconducting, can be exploited as a\nsensitive detector of various types of noise. \n\n\n\\section*{Acknowledgements}\n\nThe author appreciates collaboration and discussions with Julien Delahaye, Pertti Hakonen,\nTero Heikkil\\\"a, Rene Lindell, Mikko Paalanen, Jukka Pekola, and Mikko Sillanp\\\"a\\\"a. The author also thanks Markus\nB\\\"uttiker, Rosario Fazio, and Bertrand Reulet for interesting discussions of the results of the paper. The work was supported by \nthe Large Scale Installation Program ULTI-3 of the European Union and by the grant of the Israel Academy of Sciences and\nHumanities.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe low-energy properties of frustrated quantum spin systems -- loosely speaking,\nsystems of interacting spins in which the local energetic constraints cannot\nall be simultaneously satisfied -- have fascinated researchers for\nmany decades. These systems arise in the description of the spin degrees\nof freedom of Mott insulators, i.e. insulating states where the fluctuations of\nthe charge degrees of freedom have been suppressed by interactions\nwhile the spin degrees of freedom remain free to form non-trivial quantum\nphases. Such states are found in many materials, but can also be\nartificially made in the lab using cold atomic gases~\\cite{greiner2002,jordens2008}.\nIn most situations, the spins collectively order into some pattern\nthat can be described through a local order parameter. A more exciting possibility,\ncoined {\\it spin liquid phase}~\\cite{balents2010}, is that the spins do not order into\nsuch a local pattern; instead, a more exotic state governed by strong quantum\nfluctuations emerges.\n\nIn a famous paper in 1987, Kalmeyer and Laughlin~\\cite{kalmeyer1987}\nhypothesized a scenario where a chiral topological\nspin liquid (CSL) is formed. In this elusive phase of matter, the spins form\na collective state that can only be described in terms of emergent, non-local\ntopological properties. So far, this behavior has experimentally only been observed in fractional\nQuantum Hall systems~\\cite{tsui1982,laughlin1983}.\nSuch topologically ordered liquids~\\cite{wen1990-1} are characterized through a\nnumber of universal properties ranging from topologically protected gapless\nedge states~\\cite{halperin1982,wen1990} and a ground state degeneracies\nthat depend on the topology of the sample~\\cite{wen1990-1} to exotic\nexcitations that carry fractional charge and satisfy neither fermionic nor\nbosonic exchange statistics~\\cite{moore1991}. These \\emph{anyonic particles}~\\cite{wilczek1982}\ncan serve as key ingredients in topological quantum computers~\\cite{nayak2008},\nmaking them relevant also for technological applications.\n\nIn the specific case of the chiral spin liquid as proposed by Kalmeyer and\nLaughlin, the universal properties of the\nground state are captured by the bosonic $\\nu=1\/2$ Laughlin\nstate~\\cite{laughlin1981}. Probably the most striking property of this state\nare its semionic bulk excitations: When exchanging two such semions,\nthe wave function describing the collective state of the system acquires\na complex phase $i$, in stark contrast\nto conventional bosons or fermions, where the factor is 1 and $-1$,\nrespectively. Equally striking is the emergence of a topologically\nprotected chiral edge state with a universal spectrum at the boundary\nof the sample. This leads to unidirectional transport along the\nboundary of the sample, while the bulk remains insulating. The \ncorrespondence between edge and bulk physics has\nbeen used as a powerful experimental probe into the physics of\nfractional quantum Hall systems. Finally, the\nground state degeneracy of this state depends on the topology of the\nunderlying manifold; for example, when placed on a torus, two ground\nstates are found.\n\n\\begin{figure}\n  \\includegraphics[width=1.5in]{kagome.pdf}\n  \\includegraphics[width=1.5in]{hom.pdf}\n  \\caption{\n  {\\it Left panel:} Kagome lattice considered in this manuscript, where grey shading\n  indicates the elementary triangles. Arrows on the bonds indicate the direction\n  induced by the magnetic flux $\\Phi$ enclosed in each triangle. \n  {\\it Right panel:} Visualization of the network-model perspective on the chiral spin liquid phase arising from Hamiltonian~\\eqnref{eqn:ham}. Consistent with a chiral topological phase, a collective edge state encircles the whole systems. In this particular model, additional closed edges encircle each hexagon.\n  \\label{fig:kagome}\n  }\n\\end{figure}\n\nOver the last decades, much research has been devoted to finding\nrealistic spin Hamiltonians that have such a chiral topological phase\nas their ground state, but to this date, the only known examples are\nHamiltonians that are unlikely to be relevant for any material~\\cite{yao2007,schroter2007,tang2011,sun2011,neupert2011,nielsen2013}.\nHere, we study a simple spin model on the Kagome lattice (left panel of Fig.~\\ref{fig:kagome}) that is derived from\nthe Hubbard model, which is the minimal relevant\nmodel for itinerant interacting electrons,\nin the presence of time-reversal symmetry breaking.\nThe Hubbard Hamiltonian reads\n\\begin{equation} \\label{eqn:hubb} \\begin{split}\nH =& - \\sum_{\\langle i,j \\rangle, \\sigma} (t_{ij} c_{i \\sigma}^\\dagger c_{j \\sigma} + t_{ij}^* c_{j \\sigma}^\\dagger c_{i \\sigma}) \\\\\n& + \\frac{h_z}{2} \\sum_i (n_{i \\uparrow} - n_{i \\downarrow}) + U \\sum_{i} n_{i \\uparrow} n_{i \\downarrow}.\n\\end{split} \\end{equation}\nHere, a magnetic field induces both a Zeeman term $h_z$ as well as a flux $\\Phi$ through\neach elementary triangle of the lattice, such that for $i,j,k$ clockwise around a triangle\nwe have $t_{ij} t_{jk} t_{ki} = t^3 \\exp(i \\Phi)$, as indicated by the arrows in the left panel of\nFig.~\\ref{fig:kagome}.\nWhen $\\Phi \\neq 0$ or $h_z \\neq 0$, time-reversal symmetry is broken.\nWhen considered at half-filling, $\\langle n \\rangle = 1$, and in the limit\nof large repulsive interaction strength $U$, a Mott insulating state is formed and an effective\nspin model can be derived from perturbation theory in $t\/U$~\\cite{motrunich2006}.\n\nHere, we will demonstrate conclusively that in a very wide parameter regime where\na large enough magnetic flux $\\Phi$ breaks time-reversal symmetry,\nthe ground state of the effective spin model -- and hence also the Hubbard model in an\nappropriate parameter regime -- is a chiral topological spin liquid with\nemergent anyonic excitations.\n\n\\section{Model} \\label{sec:model}\n\nStarting from from the Hubbard model of Eqn.~\\eqnref{eqn:hubb}, a $t\/U$ expansion\nat half filling gives the following spin Hamiltonian~\\cite{motrunich2006}:\n\\begin{equation} \\label{eqn:ham} \\begin{split}\nH =&~ J_\\text{HB} \\sum_{\\langle i,j \\rangle} \\vec{S}_i \\cdot \\vec{S}_j + h_z \\sum_i S^z_i \\\\\n& +J_\\chi \\sum_{i,j,k \\in \\bigtriangleup} \\vec{S}_i \\cdot (\\vec{S}_j \\times \\vec{S}_k) + \\ldots,\n\\end{split} \\end{equation}\nwhere for the three-spin term, the $i,j,k$ are ordered clockwise around the elementary triangles of the\nKagome lattice. The term $\\chi_{ijk} = \\vec{S}_i \\cdot (\\vec{S}_j \\times \\vec{S}_k)$, referred to\nas the scalar spin chirality~\\cite{wen1989,baskaran1989},\nbreaks time-reversal symmetry and parity, but preserves SU(2) symmetry.\nTo lowest order, the coupling parameters depend on the parameters of\nthe Hubbard model as $J_\\text{HB} \\sim t^2\/U$ and $J_\\chi \\sim \\Phi t^3\/U^2$,\nignoring further subleading terms. We choose\nto parametrize the model using $J_\\text{HB} = J \\cos \\theta$ and $J_\\chi = J \\sin \\theta$\nand set $J=1$.\n\nIn the absence of time-reversal symmetry breaking ($\\theta=0$ and $h_z=0$),\nthis is the Kagome lattice nearest-neighbor Heisenberg antiferromagnet, which has become\na paradigmatic model for frustrated magnetism~\\cite{elser1989,marston1991,sachdev1992} with possible\nspin liquid ground state and relevance to the description of materials such as Herbertsmithite and\nVolborthite~\\cite{lee2007,han2012}.\nRecent numerical work~\\cite{yan2011,jiang2012-1,depenbrock2012} has indicated \nthat this model may realize a time-reversal symmetric $\\mathbb{Z}_2$ topological spin liquid,\nwhile other numerical results give evidence for a gapless spin liquid phase~\\cite{iqbal2011,clark2012}.\n\nHere, we explore the ground state phase diagram of~\\eqnref{eqn:ham} away from\nthe time-reversal invariant Heisenberg point $\\theta=0$. In particular, we find an\nextended chiral spin liquid phase around the point $\\theta=\\pi\/2$ and $h_z=0$,\nwhere the Hamiltonian reduces to the three-spin term,\n\\begin{align} \\label{eqn:csl-ham}\nH_{\\text{CSL}} &= \\sum_{i,j,k \\in \\bigtriangleup} \\chi_{ijk}.\n\\end{align}\nOur numerical results indicate that the CSL is in fact stable almost up to the\nHeisenberg point, namely for all $\\theta \\geq 0.05 \\pi$.\nWe also establish an extended range of stability against other perturbations, including\n(i) the Zeeman field,\n(ii) an easy-axis spin anisotropy in the Heisenberg term,\n(iii) a next-nearest neighbor Heisenberg term, and\n(iv) the Dzyaloshinsky-Moriya (DM) interaction induced by Rashba-type spin-orbit coupling for the fermions.\nWhile the aim of this paper is not to examine the nature of the\ntransitions out of the chiral spin liquid phase, e.g. the expected phase transition\nto the time-reversal symmetric spin liquid of the Heisenberg antiferromagnet,\nthese questions should be addressed in future work.\n\nIn the following, we will use two complementary routes to show that the ground\nstate of Eq.~\\eqnref{eqn:csl-ham} is indeed a chiral spin liquid.\nFirst, we argue for this from a powerful perspective rooted in the physics of network models of\nedge states akin to the Chalker-Coddington network model for the integer\nquantum Hall transition~\\cite{chalker1988}. We will then turn to powerful numerical\ntools to unambiguously identify the universal properties of the chiral spin liquid.\n\n\\begin{figure}\n  \\begin{tabular}{c}\n  \\includegraphics[width=3in]{puddle1.pdf} \\\\ \\hline\n  \\vspace{0.1in} \\includegraphics[width=2.5in]{puddles_2.pdf}  \n  \\end{tabular}\n  \\caption{(color online)\n  {\\it Top:} Sketch of a puddle of topological phase replacing each triangle of three spins.\n  {\\it Bottom:} Behavior of two corner-sharing triangular puddles of the topological phase. \\label{fig:puddle} }\n\\end{figure}\n\nThe key step of our first argument is to view each triangle of spins\nas the seed of a chiral topological phase, a puddle encircled by\nan edge state, as illustrated in the top panel of Fig.~\\ref{fig:puddle}. The natural\ncandidate for the phase filling the puddle is the bosonic $\\nu=1\/2$\nLaughlin state~\\cite{halperin1983}\\nocite{moore1991}, which is the simplest\nbosonic quantum Hall state known to possess the SU(2) symmetry required by our construction.\nIt is also the state envisioned by Kalmeyer and Laughlin~\\cite{kalmeyer1987}.\nForming a lattice out of the elementary triangles, we should then consider\na situation with many individual puddles of this topological phase. To\nsee what collective state is formed, we have to understand how two\ncorner-sharing triangles of the Kagome lattice are joined.\nThis situation of edges meeting at the corner spin shared by two triangles\nis an incarnation of two-channel Kondo physics~\\cite{AffleckLudwig1991,MaldacenaLudwig1997},\nfor which it is well-known that the edges will {\\it heal}~\\cite{eggert1992,kane1992}\nif the coupling\nto the center spin is symmetric, as illustrated in the lower panel\nof Fig.~\\ref{fig:puddle} and discussed in more detail in the appendix. Thus, the corner spin has\nmerged the two triangles to form a larger puddle encircled by\na single edge state, i.e. to form a larger region of the topological phase.\nWe can repeat the above step (Fig.~\\ref{fig:puddle}) for all pairs of\ncorner-sharing triangles of the Kagome lattice. The system then forms\none macroscopic, extended region of a \\emph{single} topological\nphase with one edge state encircling its outer boundary, as illustrated\nin the right panel of Fig.~\\ref{fig:kagome}, and with closed loops encircling the interior\nhexagons of the Kagome lattice. We thus obtain a direct realization of the\nKalmeyer-Laughlin state for a chiral topological spin liquid phase.\n\n\\section{Numerical identification of the CSL} \\label{sec:numerics}\n\nWe now turn to a numerical identification of the CSL \nat the chiral point $\\theta=\\pi\/2$ of Hamiltonian~\\eqnref{eqn:ham} and\nin its vicinity by studying the three\nhallmark properties of a chiral topological phase: the presence of\n(i) a gapped spectrum with a topological degeneracy on the torus,\n(ii) a gapless edge state with a universal spectrum of low-energy excitations,\nand (iii) anyonic bulk excitations.\nOn a technical level, we resort to exact diagonalization and DMRG calculations\nto extract energy spectra, entanglement spectra, and modular matrices for\nvarious system configurations.\nTo label their diameter and boundary condition, we will\nuse the notation introduced in Ref.~\\onlinecite{yan2011}; see also\nthe Methods section.\n\n\\begin{figure}\n  \\includegraphics[width=\\columnwidth]{spectrum.pdf}\n  \\caption{\n  {\\it (a)} Exact diagonalization excitation energies on small tori of type XT4-0 up to length 5. \n  {\\it (b)} Energy gap on a thin, long strip of width 4 sites; dashed lines indicate the extrapolation to $N \\rightarrow \\infty$. The two different branches denote systems with an even (black) and odd (blue) number of unit cells.\n  {\\it (c)} Entanglement entropy at the center of the same system as (b) on a semi-logarithmic scale for even (black) and odd (blue) number of unit cells; dashed lines indicate a fit. Both fits are consistent with $c=1$.\n  In all panels, $N$ is the total number of lattice sites.\n  \\label{fig:spectrum} }\n\\end{figure}\n\nWe first demonstrate that the system has a finite gap in the thermodynamic limit.\nTo this end, we consider a sequence of XT4-0 tori of length up to 5 unit cells, shown in the left panel\nof Fig.~\\ref{fig:spectrum}.\nFor systems with $N \\geq 18$ sites, there clearly is a low-lying excitation, which can be attributed\nto a two-fold near-degeneracy of the ground state. All excitations above these near-degenerate ground\nstates are separated by a spectral gap of roughly $\\Delta \\approx 0.05$.\nFurther consistent evidence for the gap can be obtained from exact diagonalization of a 36-site\nXT6-3 cluster and on XT4-2 clusters of size up to 30 sites (not shown).\nAs a further consistency check, we can extract the gap for long, thin cylinders using DMRG.\nPerforming this for cylinders of type XC4-0 with up to 100 sites,\nwe confirm that the triplet gap does not depend significantly on the length\nof the system, ruling out the presence of gapless modes propagating along the cylinder.\n\nWe conclude from this that the gap remains finite in the thermodynamic limit. The qualitative\nagreement between the spectral gap extracted for different system sizes and boundary conditions\nis also a strong indication\nthat the correlation length of the system is short compared to the system sizes we are able to\nstudy numerically. To further support this, we have calculated the spin-spin and dimer-dimer\ncorrelation functions as well as an upper bound on the asymptotic correlation\nlength on infinite cylinders. All of these indicate a correlation length on the order\nof {\\it one} unit cell. Taken together, this gives strong evidence that the properties observed on\nsmall tori and quasi-one-dimensional systems are representative of the\ntwo-dimensional phase in the thermodynamic limit.\n\nThe observed two-fold near-degeneracy is consistent with what is\nexpected for the CSL, namely a\n$2^g$-fold ground state degeneracy on a manifold of genus $g$, which\nwill be split by an amount that is exponentially small in the system size.\nWe also find two states $\\ket{\\Psi_a}$ with very similar energy densities for infinite\ncylinders of type XC8-4 and XC12-6. As explained in the\nMethods section, the two states can be identified by\ntheir well-defined topological flux $a$ through the cylinder,\nwhich for the $\\nu=1\/2$ Laughlin state can be the identity\n($a = \\mathds{1}$) or a semion ($a=\\mathrm{s}$).\n\n\\subhead{Edge physics}\nPlacing a chiral topological phase on a cylinder or disk, a gapless chiral edge state emerges with\na universal spectrum governed by a conformal field theory~\\cite{halperin1982,wen1990}.\nTo observe this, we consider the spectral gap of the system\nwhen placed on a thin, long strip with a fixed width of 4 sites.\nIn stark contrast with the case of a thin long cylinder, the spin gap\nvanishes as $a\/L+b\/L^2$, where $a$ and $b$ are\nparameters of the fit (Fig.~\\ref{fig:spectrum}(b)).\nThis is a hallmark signature of a gapless edge mode. We can further\npinpoint the universality class of the edge theory by extracting its central charge $c$\nfrom the entanglement entropy. As shown in the Fig.~\\ref{fig:spectrum}(c), we find good agreement\nwith a value of $c=1$, which is precisely that expected for the chiral SU(2)$_1$ Wess-Zumino-Witten\nconformal field theory describing the edge of a $\\nu=1\/2$ Laughlin state.\n\n\\begin{figure}\n  \\includegraphics[width=\\columnwidth]{cft.pdf}\n  \\caption{Entanglement spectrum of the reduced density matrix $\\rho_a$ for one half of an infinite cylinder obtained for both ground states $\\ket{\\Psi_\\mathds{1}}$ (left) and $\\ket{\\Psi_\\mathrm{s}}$ (right panel). The entanglement energies shown on the vertical axis, up to the global shift and rescaling, are given by $E_{a,\\sigma} = -\\log (p_{a,\\sigma})$, where $p_{a,\\sigma}$ are the eigenvalues of $\\rho_a$. The horizontal axis shows the momentum in the transverse direction of the corresponding eigenvectors of $\\rho_a$. Each tower is identified by its $S^z$ quantum number as indicated by the blue label; we have offset the momentum of different towers by $2\\pi$ to improve clarity. The cylinder used here is XC12-6.\n }\n \\label{fig:cft}\n\\end{figure}\n\nAs an even more refined probe, we use the entanglement spectrum, which\nreflects the same universal properties as the physical edge spectrum~\\cite{li2008,qi2012,chandran2011,dubail2012,swingle2012}.\nFor each of the two ground states $\\ket{\\Psi_a}$ obtained for an infinite cylinder,\nthe entanglement spectrum, see Fig.~\\ref{fig:cft}, is consistent with\nthe corresponding sector of the\nchiral SU$(2)_1$ Wess-Zumino-Witten conformal field theory:\nThe entanglement spectrum of $\\ket{\\Psi_\\mathds{1}}$ displays precisely the sequence\nof degeneracies of the tower of Kac-Moody descendants\nof the identity primary field (1-1-2-3-5-...). These are reproduced by counting\nthe number of low-lying close-by states in each tower grouped by momentum and spin quantum numbers.\nSimilarly, the entanglement spectrum of $\\ket{\\Psi_\\mathrm{s}}$ displays the degeneracies of\nthe spin-1\/2 primary field and its descendants (also 1-1-2-3-5-...).\nWe note that in the identity sector, all towers carry integer representations of the spin quantum number, whereas in the semion\nsector they carry half-integer representations. In both ground states, the levels can be grouped into\nSU(2) multipletts.\n\n\\subhead{Emergent anyons}\nThe bulk of the chiral spin liquid phase has anyonic excitations, referred to as semions.\nThe topological properties of these quasiparticles can be characterized through their modular\n$T$ and $S$ matrices~\\cite{nayak2008}. The $T$ matrix contains the\ncentral charge $c$ and the self-statistics of the anyonic particles, i.e. the phase\nthat is obtained when two particles of the same kind are exchanged.\nThe $S$ matrix contains the mutual statistics of the anyonic quasiparticles, their\nquantum dimensions (counting the internal degrees of freedom\nof each particle), and the total quantum dimension of the phase. More detailed\ndefinitions of these quantities are given in the Methods summary.\n\nFor a fixed number of quasiparticles, only a finite number of\npossible $S$ and $T$ matrices exist~\\cite{rowell2009,bruillard2013}. For two types of\nquasiparticles (as in the case of the $\\nu=1\/2$ bosonic Laughlin state) only two choices are possible~\\cite{rowell2009}.\nTherefore, by numerically calculating the $S$ and $T$ matrices and\ncomparing them against the two possibilities, we have fully identified\nthe universal properties of the topological phase.\nFor the $\\nu=1\/2$ Laughlin state, the modular matrices are\n\\begin{align} \\label{TS}\nT&=e^{-i \\frac{2\\pi}{24}} \\left[\\begin{matrix} 1 & 0 \\\\ 0 & i \\end{matrix} \\right]\n&S&=\\frac{1}{\\sqrt{2}}\\left[\\begin{matrix} 1 & 1 \\\\ 1 & -1 \\end{matrix}\\right].\n\\end{align}\n\nFor an XT8-4 torus of $48$ sites at $\\theta=0.05 \\pi$, where the\nfinite-size corrections to this quantity are minimal, we obtain\n\\begin{eqnarray}\nT &=& e^{-i \\frac{2\\pi}{24} 0.988} \\left[\n\\begin{matrix}\n1 & 0 \\\\\n0 & i \\cdot e^{-i0.0021 \\pi}\n\\end{matrix}\n\\right] \\ , \\label{numT} \\label{numS} \\\\\nS &=& \\frac{1}{\\sqrt{2}} \\left[\n\\begin{matrix}\n0.996 & 0.995 \\\\\n0.996 & -0.994 e^{-i 0.0019 \\pi}\n\\end{matrix} \\right] \\ . \\nonumber\n\\end{eqnarray}\nThis is in very good agreement with the $T$ and $S$ matrices for the $\\nu=1\/2$ Laughlin state\ngiven in Eqn.~\\eqnref{TS} and provides the strongest confirmation of the nature of the\nbulk topological phase.\nThe correct normalization of the first row or column of the $S$ matrix\nindicates that we have indeed obtained a full set of ground states.\nWe can also read off the total quantum dimension $\\mathcal{D}=1\/S_{\\mathds{1} \\mathds{1}}=\\sqrt{2}\/0.996$ of the phase,\nwhich determines the topological entanglement entropy~\\cite{kitaev2006-tee,levin2006}\nthat has been widely used to identify topological phases.\nFurthermore, the central charge $c=0.988$ is in excellent agreement with the prediction\nand the value extracted from the edge above.\n\n\\begin{figure}\n  \\includegraphics[width=\\columnwidth]{pd.pdf}\n  \\caption{\n  Singlet and triplet gaps as a function of $\\theta$ for an infinite XC8-4 cylinder.\n  The triplet gap is a lower bound on the critical magnetic field $h_c$; hence, the shaded region in\n  the middle panel indicates the minimal stability of the phase in the $\\theta$-$h_z$ phase diagram.\n  \\label{fig:pd} }\n\\end{figure}\n\n\\subhead{Stability of the chiral spin liquid}\nTo establish the region in which the phase persists as $\\theta$ is tuned in the range $\\theta \\in [0,\\pi\/2]$,\nwe first consider the fidelity~\\cite{zanardi2006} $F(\\theta) = \\langle \\Psi_a(\\theta-\\epsilon) | \\Psi_a(\\theta+\\epsilon) \\rangle$\nshown in the bottom panel of Fig.~\\ref{fig:pd} (for the precise definition of this quantity for infinite\nsystems, see the appendix). The fidelity remains near unity as $\\theta$ is tuned away from\nthe chiral point $\\theta=\\pi\/2$, until it suddenly drops for $\\theta < 0.05 \\pi$, indicating a transition.\nFurther support for the extended stability of the CSL is found in various other characteristics,\nincluding the spectral gap, the modular matrices and the entanglement spectrum.\nThis is remarkable as it indicates that tuning away from the Heisenberg model ($\\theta=0$)\nwith a small critical chiral coupling of $(J_\\chi\/J_\\text{HB})_\\text{crit} \\leq \\tan(0.05 \\pi) \\approx 0.16$\nis sufficient to drive the system into the chiral phase. \n\nIn experimental scenarios, a Zeeman magnetic field $h_z$ is likely to be generated along with\nthe orbital magnetic field that induces the three-spin chiral term. The relative strength of the orbital\nmagnetic field and the Zeeman field is determined by the $g$-factor and the ratio $t\/U$.\nThe energy gap in the triplet sector gives a lower bound on the critical field strength $h_c$ up \nto which the CSL phase is stable. The values for the triplet gap shown in the middle panel of Fig.~\\ref{fig:pd}\nremain large all the way from the fully chiral point ($\\theta=\\pi\/2$) to the transition out of the\nCSL towards the Heisenberg point ($\\theta=0$).\n\nIn the top panel of Fig.~\\ref{fig:pd}, we also show the singlet gap, i.e. the gap to the lowest excitation\nin the $S_z=0$ sector. As opposed to the triplet gap, which appears to remain large across the\ntransition, the singlet gap decreases as the transition out of the CSL is approached. Note that the gaps may\nbe rounded off at the transition by effects due to either the finite diameter or the finite bond dimension\nof the matrix-product state ansatz.\nWe point out, however, that a closing only of the singlet and not the triplet\ngap is consistent with the scenario for the transition from the chiral spin liquid into a\ndoubled semion phase (twisted $\\mathbb{Z}_2$ topological phase) studied in Ref.~\\onlinecite{barkeshli2013}.\n\n\\section{Outlook}\n\nWe have taken an important step towards finding realistic models for a chiral spin\nliquid in a frustrated spin system. \nWe believe that this will nucleate new research efforts both by theorists and\nexperimentalists. From a theoretical point of view, studying the transition from\nthe chiral spin liquid to the putative time-reversal symmetric spin liquid in the\nHeisenberg model will provide the unique opportunity to study a topological\nphase transition in a realistic model, and may provide invaluable insights into\nthe physics of frustrated spins on the Kagome lattice. For experimentalists, our\nwork will provide a guide in searching for realizations of bosonic fractional\nQuantum Hall physics in the lab, be it in materials that have Kagome lattice\nstructure and form Mott insulators, or by engineering such systems in cold atomic gases.\n\n \\vspace{0.15in}\n{\\small During completion of this work, we became aware of related work in Refs.~\\onlinecite{he2013,gong2013}.}\n\n\\acknowledgements\nThe DMRG code was developed with support from the Swiss Platform for High-Performance\nand High-Productivity Computing (HP2C) and based on the ALPS libraries~\\cite{bauer2011-alps}.\nS.T. was supported, in part, by  SFB TR 12 of the DFG. A.W.W.L was supported, in part, by\nNSF DMR-1309667. We acknowledge illuminating discussions with participants of the KITP\nworkshop \\emph{Frustrated Magnetism and Quantum Spin Liquids} (Fall 2012) as well as\nM. Barkeshli and P. Bonderson. This research was supported in part by Perimeter Institute for\nTheoretical Physics. Research at Perimeter Institute is supported by the Government of Canada\nthrough Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}