diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzmsur" "b/data_all_eng_slimpj/shuffled/split2/finalzzmsur" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzmsur" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nPlanet formation occurs in disks around young stellar objects. Interactions between planets and disks are very complex. Young planets are expected to cause rings, cavities, spirals, and disturbances in the velocity field and other features in the disk, which in turn may be used to infer the presence of these young planets. In the past few years, much evidence about this phase of planet formation has been accumulated because high-resolution images in the millimeter and sub-millimeter wavelength ranges have been provided by the Very Large Array (VLA) and the Atacama Large Millimeter Array (ALMA) (see e.g. the case of HL Tau: \\cite{ALMA2015}), and by high-contrast imagers such as the Gemini Planet Imager (GPI: \\cite{Macintosh2014}) and SPHERE (Spectro-\nPolarimetic High contrast imager for Exoplanets REsearch: (\\cite{Beuzit2008}; see, e.g., \\cite{Avenhaus2018}). The literature on indirect evidence of the presence of planets is now becoming very rich, and nearby young stars surrounded by gas-rich disks are intensively studied for this purpose. In most cases, available data cannot fully eliminate alternative hypotheses, or the data have ambiguous interpretations (see, e.g., \\cite{Bae2018} and \\cite{Dong2018}), although strong indirect evidence of the presence of planets from local disturbances of the velocity field have recently been considered for the case of HD~163296 (\\cite{Pinte2018, Teague2018}). In general, small grains are thought to be more strongly coupled with gas and are thus less sensitive to radial drift and concentration that can strongly affect large grains (see the discussion in \\cite{Dipierro2018}). For this reason, observations at short wavelengths provide an important complementary view of what can be seen with ALMA. On the other hand, a direct detection of still-forming planets embedded within primordial gas-rich disks, which is expected to be possible with high-contrast imaging in the near infrared (NIR), is still scarce; remarkable cases are LkCa-15 (\\cite{Kraus2012, Sallum2015}) and PDS-70 (\\cite{Keppler2018, Muller2018, Wagner2018}). In particular, in this second case, a clear detection of an accreting planet in the cavity between the inner and outer ring was obtained, making it an archetype for planet formation and planet-disk interactions. However, many cases remain ambiguous; a classical example is HD~100546 (see, e.g., \\cite{Quanz2013a, Currie2014, Quanz2015, Currie2015, Rameau2017, Sissa2018}).\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=18.8cm]{qphi.jpeg}\n \n \n\\caption{View of the surroundings of HD~169142 obtained from polarimetric observations: the left panel shows the Q$_\\Phi$\\ image in the J band acquired with SPHERE (\\cite{Pohl2017}) on a linear scale. The two rings are clearly visible. The right panel shows a pseudo-ADI image of the inner regions obtained by differentiating the Q$_\\Phi$\\ image (see \\cite{Ligi2017}, for more details). The white cross marks the position of the star, and the cyan circle shows the size of the coronagraphic mask. The other labels refer to the blobs we discuss in this paper that are also visible in these images. The color scale of the differential image is five times less extended to show the faint structures better. In both panels, N is up and E to the left; a segment represents 1 and 0.5 arcsec in the left and right panel, respectively. }\n\\label{f:pdi}\n\\end{figure*}\n\nHD~169142 is a very young Herbig Ae-Be star with a mass of 1.65-2~M$_\\odot$ and an age of 5-11~Myr (\\cite{Blondel2006, Manoj2007}) that is surrounded by a gas-rich disk ($i=13$~degree; \\cite{Raman2006}; $PA=5$~degree; \\cite{Fedele2017}) that is seen almost face-on. The parallax is $8.77\\pm 0.06$~mas (GAIA DR2 2018). Disk structures dominate the inner regions around HD~169142 (see, e.g., \\cite{Ligi2017}). Figure~\\ref{f:pdi} shows the view obtained from polarimetric observations: the left panel shows the Q$_\\Phi$\\ image in the J band obtained by \\cite{Pohl2017} using SPHERE on a linear scale, and the two rings are clearly visible. The right panel shows a pseudo-ADI image of the inner regions obtained by differentiating the Q$_\\Phi$\\ image (see \\cite{Ligi2017}, for more details). \\cite{Biller2014} and \\cite{Reggiani2014} discussed the possible presence of a point source candidate at small separation ($<0.2$~arcsec from the star). However, the analysis by Ligi et al. based on SPHERE data does not support or refute these claims; in particular, they suggested that the candidate identified by Biller et al. might be a disk feature rather than a planet. Polarimetric images with the adaptive optics system NACO at the Very Large Telescope (VLT) (\\cite{Quanz2013b}), SPHERE (\\cite{Pohl2017, Bertrang2018}) and GPI (\\cite{Monnier2017}) show a gap at around 36~au, with an outer ring at a separation $>$40~au from the star. This agrees very well with the position of the rings obtained from ALMA data (\\cite{Fedele2017}); similar results were obtained from VLA data (\\cite{Osorio2014, Macias2017}). We summarize this information about the disk structure in Table~\\ref{t:rings} and call the ring at 0.17-0.28 arcsec from the star Ring 1 and the ring at 0.48-0.64 arcsec Ring 2. We remark that in addition to these two rings, both the spectral energy distribution (\\cite{Wagner2015}) and interferometric observations (\\cite{Lazareff2017, Chen2018}) show an inner disk at a separation smaller than 3 au. This inner disk is unresolved from the star in high-contrast images and consistent with ongoing accretion from it onto the young central star. While the cavities between the rings seem devoid of small dust, some gas is present there (\\cite{Osorio2014, Macias2017, Fedele2017}). \\cite{Fedele2017} and \\cite{Bertrang2018} have suggested the possibility that the gap between Rings 1 and 2 is caused by a planet with a mass slightly higher than that of Jupiter. However, this planet has not yet been observed, possibly because it is at the limit of or beyond current capabilities of high-contrast imagers. On the other hand, \\cite{Bertrang2018} found a radial gap in Ring 1 at PA$\\sim 50$\\ degree that might correspond to a similar radial gap found by \\cite{Quanz2013b} at PA$\\sim 80$\\ degree. The authors noted that if this correspondence were real, then this gap might be caused by a planet at about 0.14 arcsec from the star. So far, this planet has not been unambiguously detected either.\n\n\\begin{table*}\n\\centering\n\\caption{Rings around HD169142 from the literature}\n\\begin{tabular}{lccc}\n\\hline\n\\hline\nInstrument & Source & Ring 1 & Ring 2 \\\\\n & & arcsec & arcsec \\\\\n\\hline\nALMA & \\cite{Fedele2017} & 0.17-0.28 & 0.48-0.64 \\\\\nVLA & \\cite{Osorio2014} & 0.17-0.28 & 0.48- \\\\\nSUBARU-COMICS & \\cite{Honda2012} & 0.16- & \\\\\nNACO & \\cite{Quanz2013b} & 0.17-0.27 & 0.48-0.55 \\\\\nSPHERE-ZIMPOL & \\cite{Bertrang2018} & 0.18-0.25 & 0.47-0.63 \\\\\nSPHERE-IRDIS & \\cite{Pohl2017} & 0.14-0.22 & 0.48-0.64 \\\\\nGPI & \\cite{Monnier2017} & 0.18 & 0.51 \\\\\n\\hline\n\\end{tabular}\n\\label{t:rings}\n\\end{table*}\n\nIn this paper, we pursue a new view on the subject through analyzing high-contrast images. In particular, we underline that while polarimetric observations in the NIR and millimeter observations are best to reveal the overall structure of the disk, pupil-stabilized NIR observations where angular differential imaging can be applied may reveal fainter structures on a smaller scale. The risk of false alarms inherent to the image-processing procedures used in high-contrast imaging can be mitigated by comparing different sets of observations taken at intervals of months or years. In the case of HD~169142, this is exemplified by the study of \\cite{Ligi2017}, who identified a number of blobs within Ring 1. We have now accumulated a quite consistent series of observations of this star with SPHERE that extends the set of data considered by Ligi et al. The observations have a comparable limiting contrast so that we may try to combine this whole data set to improve our knowledge of this system. The combination of different data sets acquired over a few years offers several advantages. In addition to verification of previous claims, we might try to detect persistent features around HD~169142 using a coincidence method to obtain a combined image that is deeper than the individual images and allows a quantitative discussion of the false-alarm probability of detected features. The expected orbital motion needs to be taken into account in this.\n\nIn Section 2 we describe observation and analysis methods. In Section 3 we present the main results about the blobs we detect around the star. in Section 4 we discuss the spiral arms within the disk and the possible connection to the blobs. Conclusions are given in Section 5.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=18.8truecm]{all_epochs.jpeg}\n \n \n\\caption{Signal-to-noise ratio maps for the individual epochs for IFS. The individual S\/N maps are obtained from the ASDI PCA algorithm using 50 components and making a median over the wavelength (see \\cite{Mesa2015}). In the upper row we show in the left panel JD=57180.17, in the middle panel JD=57201.12, and in the right panel JD=57499.34. In the lower row we show in the left panel JD=57566.15, in the middle panel JD=57873.30, and in the right panel JD=58288.19. In all panels, the central 0.1 arcsec is masked, the solid white line at the bottom represents 1 arcsec, the white cross represents the position of the star. N is up and E to the left.}\n\\label{f:individual}\n\\end{figure*}\n\n\n\\section{Observation and data analysis}\n\nData were acquired with the SPHERE high-contrast imager (\\cite{Beuzit2008}) at the ESO VLT Unit Telescope 3 within the guaranteed time observations used for the SHINE (SpHere INfrared survey for Exoplanets) survey (\\cite{Chauvin2017}). Data acquired up to 2017 have been described in \\cite{Ligi2017}. Here we add new data acquired in 2018 and study the system anew using different ways to combine different images. In these observations, we used SPHERE with both the Integral Field Spectrograph (IFS : \\cite{Claudi2008}) and the Infra-Red Dual Imaging and Spectrograph (IRDIS: \\cite{Dohlen2008, Vigan2010} simultaneously. IFS was used in two modes: Y-J, that is, with spectra from 0.95 to 1.35~$\\mu$m and a resolution of $R\\sim 50$; and Y-H, with spectra from 0.95 to 1.65~$\\mu$m and a resolution of $R\\sim 30$. When IFS was in Y-J mode, IRDIS observed in the H2-H3 narrow bands (1.59 and 1.66~$\\mu$m, respectively); when IFS was in Y-H mode, IRDIS observed in K1-K2 bands (2.09 and 2.25~$\\mu$m, respectively). Hereafter we mainly consider data acquired with the IFS; IRDIS data are considered for the photometry in the K1-K2 bands. We considered the six best observations obtained for HD169142 (see Table~\\ref{t:obs}). Most of the epochs were obtained with an apodized Lyot coronagraph (\\cite{Boccaletti2008}: the field mask in YJH has a radius of 92~mas, and of 120~mas for the K-band coronagraph). Two of the observations (obtained in better observing conditions) were acquired without the coronagraph in order to study the very central region around the star. The use of the coronagraph allows a better contrast at separation larger than $\\sim 0.1$~arcsec. For all data sets, the observations were acquired in pupil-stabilized mode. In addition to the science data, we acquired three kind of on-sky calibrations: (i) a flux calibration obtained by offsetting the star position by about 0.5 arcsec, that is, out of the coronagraphic mask (point spread function, PSF, calibration). (ii) An image acquired by imprinting a bidimensional sinusoidal pattern (waffle calibration) on the deformable mirror. The symmetric replicas of the stellar images obtained by this second calibration allow an accurate determination of the star centers even when the coronagraphic field mask is in place. These calibrations were obtained both before and after the science observation, and the results were averaged. (iii) Finally, an empty field was observed at the end of the whole sequence to allow proper sky subtraction. This is relevant in particular for the K2 data sets.\n\nData were reduced to a 4D datacube (x, y, time, and $\\lambda$) at the SPHERE Data Center in Grenoble (\\cite{Delorme2017}) using the standard procedures in the SPHERE pipeline (DRH: \\cite{Pavlov2008}) and special routines that recenter individual images using the satellite spot calibration, and correct for anamorphism, true north, and filter transmission. Faint structures can be detected in these images using differential imaging. Various differential imaging procedures were run on these data sets. We used here results obtained with a principal component analysis (PCA; see \\cite{Soummer2012}) applied to the whole 4D datacubes, which combines both angular and spectral differential imaging in a single step (ASDI-PCA: see \\cite{Mesa2015}). The PCA algorithm we used is the singular-value decomposition that generates the eigenvectors and eigenvalues that are used to reconstruct the original data. A principal components subset was used to generate an image with the quasi-static noise pattern that can then be subtracted from the original image. Clearly, the larger the number of principal components, the better the noise subtraction, but this also means that the signal from possible faint companion objects is cancelled out more strongly. Most of the results were obtained using 50 modes, but we also considered other numbers of modes (10, 25, 100, and 150 modes). To avoid spectrum distortion characteristics of the ASDI-PCA, photometry was obtained using a monochromatic PCA with only two modes for each spectral channel. Photometry was obtained with respect to the maximum of the PSF calibration corrected for the attenuation inherent to the PCA. The final step of the procedure was to obtain signal-to-noise ratio (S\/N) maps from the IFS images obtained by making a median over wavelengths. \n\nFinally, we also used the $Q_\\Phi$\\ image obtained by \\cite{Pohl2017} for astrometry, reduced as described in that paper and in \\cite{Ligi2017}. We note that this data set was obtained with the YJ field mask, whose radius is 72.5~mas.\n\n\\begin{table*}\n\\caption{Journal of observations}\n\\begin{centering}\n\\begin{tabular}{lccccccl}\n\\hline\n\\hline\nJD & Mode & nDIT$\\times$DIT & Angle & Seeing & lim. cont & coro & Ref\\\\\n & & (sec) & (degree) & (arcsec) & (mag) & &\\\\\n\\hline\n57145. & Pol J & 3180 & Field & 0.90 & & YJ & \\cite{Pohl2017} \\\\\n57180.17 & Y-J & 86$\\times$64 & ~45.82 & 1.57 & 13.13 & YJH & \\cite{Ligi2017}\\\\ \n57201.12 & Y-H & 65$\\times$64 & ~36.42 & 1.00 & 13.52 & YJH & \\cite{Ligi2017}\\\\ \n57499.34 & Y-J & 77$\\times$64 & 144.62 & 1.88 & 13.06 & YJH & \\cite{Ligi2017}\\\\ \n57566.15 & Y-H & 322$\\times$2 & 147.33 & 0.67 & 13.64 & no & \\cite{Ligi2017}\\\\ \n57873.30 & Y-H & 192$\\times$2 & ~98.82 & 0.62 & 14.07 & no & \\cite{Ligi2017}\\\\\n58288.19 & Y-H & 48$\\times$96 & 120.17 & 1.19 & 13.79 & K & This paper\\\\ \n\\hline\n\\end{tabular}\n\\end{centering}\n\\label{t:obs}\n\\end{table*}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=18.8cm]{mixed.jpeg}\n \n \n\\caption{Upper left panel: Median over time of the wavelength-collapsed images of HD169142 obtained with an ADI PCA algorithm, one mode per wavelength. Upper right panel: Image obtained by averaging the S\/N maps for the individual epochs for IFS. The individual S\/N maps are obtained from the ASDI PCA algorithm using 50 components and making a median of the wavelength (see \\cite{Mesa2015}), and they were rotated for Keplerian motion to the last image before making the median. Lower panel: Coincidence image obtained from the same data set. In all panels, the solid line at the bottom of each panel represents 1 arcsec, and a white cross shows the position of the star. N is up and E to the left.}\n\\label{f:sum}\n\\end{figure*}\n\n\\section{Results}\n\nThe following discussion is based on the application of differential imaging algorithms that allow detecting faint structure that is not otherwise easily detectable in the images. The typical contrast of Ring 1 that we were able to measure using simple subtraction of a reference image is about $1.5\\times 10^{-3}$. The structures we consider in this paper are more than an order of magnitude fainter. They represent small fluctuations of the signal that cannot be detected without differential imaging.\n\nFigure~\\ref{f:individual} shows the S\/N maps obtained by applying the PCA ASDI algorithm to the IFS data for the individual epochs. The images have a linear scale from S\/N=0 (dark) to S\/N=5 (bright). These figures clearly show a similar pattern of bright spots, as well as a rotation of these spots with time. This suggests that a combination of the images that takes into account a Keplerian motion around the star should improve detection of the real pattern present in the data. The full solution is quite complex, leaving many free parameters, and may be attempted using an approach such as that considered by K-stacker (\\cite{Nowak2018}). However, a simplified approach that greatly reduces the number of free parameters is to assume that the system is seen face-on and that the orbits are circular: if the distance is known, the only free parameter is the stellar mass. This appears to be a reasonable approximation for disk-related features around HD~169142 because in this case, we only consider a fraction of the orbit. On the other hand, observations spread over a few years enable separating static features that are due to radiative transfer effects from scattered-light fluctuations that are due to moving clouds or sub-stellar objects.\n\nThe upper panels of Figure~\\ref{f:sum} show images of HD169142 obtained by combining the six individual images, assuming the distance given by GAIA DR2, a mass of 1.7~M$_\\odot$\\ (see below), and circular orbits.\n\n\\subsection{Coincidence images}\n\nTo improve our ability of discerning faint signals, we combined data from different epochs using a coincidence map (see the lower panel of Figure~\\ref{f:sum}). The principle of this coincidence map is to start with S\/N maps for individual epochs. We used S\/N maps after correcting for the small-number-statistics effect using the formula by \\cite{Mawet2014}. The maps were then multiplied by each other. The S\/N maps average to zero, with both positive and negative values for individual pixels. Of course, this may result in a false-positive signal for an even number of negative signals in the individual S\/N maps. To avoid this problem, we arbitrarily set to negative the result for a given pixel when the signal for that pixel was negative for at least one epoch. Of course, this is not a realistic flux map: the aim is merely to identify consistent signals throughout all individual images.\n\nTo consider the orbital motion around the star over the three years covered by our observations, we divided the field into 65 rings, each one 2 IFS pixels wide (15 mas, i.e., about 1.8 au at the distance of HD~169142). For each ring, we rotated the S\/N maps obtained at different epochs with respect to the first reference image according to Kepler's third law. When the distance to the star was fixed, the only free parameter remaining in this model is the (dynamical) stellar mass. If there is a companion orbiting the star, the signal is maximized for a value of the mass that, if the assumptions made (circular motion seen face on) are correct, is the dynamical mass of the star. If these hypotheses are not correct, the estimate of the mass is incorrect, by a value that depends on the real orbital parameters. We adopted a mass of 1.7~$M_\\odot$, the GAIA parallax, and clockwise rotation, as indicated by the analysis of motion of disk features in \\cite{Ligi2017}; see also \\cite{Macias2017}. Figure~\\ref{f:sum} shows the coincidence map and a mean of the S\/N maps for the six epochs for this value of the mass.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{compdisk.jpeg}\n \n \n\\caption{Same as the lower panel of Fig~\\ref{f:sum}, but showing the edges of the two disk rings (Ring 1 in green, Ring 2 in magenta). The ring edges are drawn according to \\cite{Fedele2017}. The blobs are labeled. The white solid line represents 1 arcsec, and the white cross shows the position of the star. N is up and E to the left.}\n\\label{f:rings}\n\\end{figure}\n\n\\subsection{Blob detection}\n\nA quite large number of blobs can be found around HD~169142. Several of them are found consistently in all individual images and are also visible in the J-band $Q_\\Phi$ image seen in Figure~\\ref{f:pdi}; some of them have been identified and discussed by \\cite{Ligi2017}. We fixed our attention on four of them (see Figure~\\ref{f:pdi} and Figure~\\ref{f:rings} for their definition). The two brightest blobs (B and C) are within Ring 1 and have been identified by \\cite{Ligi2017}; they called them blobs A and B, respectively. Our blob A is closer to the star than Ring 1. Blob D is between Rings 1 and 2. All of these blobs appear to be slightly extended. We verified in the individual images that this is not an artifact caused by combining individual images. For instance, when we consider the best set of data (the last set from June 2018), the FWHM of blobs B and C can be measured with reasonable accuracy at about 40 mas, which is significantly larger than expected for a point source at this separation (about 26 mas, after applying differential imaging). To better estimate the physical size of the blobs, we compared the FWHM measured in our differential images with the FWHMs obtained for fake blobs that are the result of convolving Gaussian profiles with the observed PSF inserted into the images at the same separation but at a different position angle, and processed through the same differential imaging procedure. We repeated this procedure for the images obtained considering 50 modes (best image for detection) and with a less aggressive image where only 25 modes were considered, which better conserves the original shape of the blobs. In this way, we found that the FWHM of blobs B and C is the same as that of fake Gaussian blobs with an intrinsic FWHM of 42 and 30 mas for blobs B and C, respectively. However, these are average values for tangential and radial profiles (with respect to the star): both blobs appear elongated in the tangential (rotation) direction with axis ratios of 1.4 (blob B) and 1.9 (blob C)\\footnote{This is not as obvious from a simple visual inspection of the images because the ADI processing that is implicit in the PCA-ASDI procedure we used deforms the images.}. The uncertainty on this size estimate is of about 7 mas, as obtained by comparing the results obtained in individual images. This size corresponds to $\\sim 4-5$~au, with an uncertainty of about 1~au. This result should be considered with some caution because the light distribution of the blobs might be not well reproduced by Gaussians.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{zoomD.jpeg}\n \n \n\\caption{Zoom of a two-color image of the region around blob D. This image was constructed using the K2 observation of JD=2458288.19 (red) and the weighted sum of all the IFS images (collapsed against wavelength) and rotated for a Keplerian motion assuming that the star has a mass of 1.7~M$_\\odot$ (blue). This last image is for the same epoch as the K2 observation. For clarity, the region within 0.28 arcsec from the star (i.e., within the outer edge of Ring 1) was masked in the K2-band image. The green circle is centered on the position of the blob measured in the K2 image. We note the different aspect and small offset between the position of the blob in the K2 image with respect to that at shorter wavelength. The white solid line represents 1 arcsec. N is up and E to the left. The white cross marks the position of the star.}\n\\label{f:blobd}\n\\end{figure}\n\nIn Figure~\\ref{f:blobd} we show a zoom of the region around blob D in a two-color image. The blue structures visible in the image are obtained through IFS in the Y, J, and H bands. They could be interpreted as stellar light scattered by a (dusty) spiral structure around a protoplanet that is accreting material funnelled through the spiral arm from the disk. This interpretation agrees with the detection in the $Q_\\Phi$\\ image (see Figure~\\ref{f:pdi}). In the same image, the red structures are obtained through IRDIS observing in the K2 band. In particular, the structure in the green circle that appears to be much more similar to a point source might indicate a planetary photosphere. The position of the blob in the K2 image is Sep=332 mas, PA=34.9 degree, which is not the photocenter at shorter wavelengths. Even if this interpretation is speculative (there are other structures in this image that we consider as noise), various circumstantial arguments discussed below possibly support it. We return to this point in the next section.\n\nThere is of course some probability that these detections are spurious. In order to estimate the false-alarm probability (FAP), we proceeded as follows. First, we fixed the stellar mass at the value given by fitting isochrones (1.7~$M_\\odot$). With this assumption, the prediction for the orbital motion is fully independent of the SHINE data set. We derotated the individual images to the same epoch using the same approach as described above (ring by ring). We searched for signals in the final coincidence data set using the FIND procedure in IDL. We recovered the detection of the candidate. We ordered the different epochs according to the value of the S\/N at the candidate positions (separately for each candidate). We then used binomial statistics on the remaining epochs (i.e., excluding the reference with the highest S\/N), considering as number of trials the number of pixels with an S\/N higher than the S\/N measured in the candidate position in the image giving the highest S\/N value at this position. To estimate the probability in the binomial statistics, we considered the product $\\prod_c$\\ of the S\/N rankings in the pixel corresponding to the candidate position in the remaining images, and compared this product to a similar product $\\prod_r$ obtained from random extractions. We repeated the random extraction $10^7$ times, and assumed that the probability of success is given by the fraction of cases where $\\prod_r < \\prod_c$.\n\nWith this approach, we obtained the FAP values listed in the second column of Table~\\ref{t:blobrot}; values for blobs B, C, and D are highly significant.\n\n\n\\subsection{Blob astrometry}\n\n\\begin{figure*}\n\\begin{tabular}{cc}\n\\centering\n\\includegraphics[width=8truecm]{astro_bloba.png}&\n\\includegraphics[width=8truecm]{astro_blobb.png}\\\\\n\\includegraphics[width=8truecm]{astro_blobc.png}&\n\\includegraphics[width=8truecm]{astro_blobd.png}\\\\\n\\end{tabular}\n \n \n\\caption{Variation in PA of the blobs with time. Upper left panel: blob A. Upper right panel: blob B. Lower left panel: blob C. Lower right panel: blob D. The dashed lines are best-fit lines through the points. Dash-dotted lines are predictions for circular orbits assuming a mass of 1.85~M$_\\odot$\\ for the star.}\n\\label{f:astro}\n\\end{figure*}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{kepler_motion.png}\n \\caption{Run of the angular speed as a function of separation for the four blobs around HD169142; these values are on the disk plane. Overimposed we show predictions for circular Keplerian motion for three different values for the stellar mass (1.5, 2.0, and 2.5 M$_\\odot$).}\n \\label{f:kepler}\n\\end{figure}\n\nAll these blobs rotate around the star. This can be shown by measuring their position in the individual images (see Table~\\ref{t:blobastro}). We used the IDL FIND algorithm that uses marginal Gaussian distributions to measure the position of the spot centers in the ASDI 50 components images obtained at the various epochs. In addition, we also measured the blob positions in the polarimetric image. We found that the rotational speed decreases with separation (see Table~\\ref{t:blobrot} and Figure~\\ref{f:astro}), as expected for Keplerian motions. Figure~\\ref{f:kepler} shows the run of the angular speed as a function of separation for the four blobs around HD169142; these values are on the disk plane. Overimposed are predictions for circular Keplerian motion for three different values for the mass of the star (1.5, 2.0, and 2.5 M$_\\odot$). \n\nWhen we interpret the observed angular motion as Keplerian circular orbits in the disk plane, we can determine the mass of HD~169142. When we use the three blobs B, C, and D, the mass of the star is $1.85\\pm 0.25$~M$_\\odot$ (we did not use blob A here because it has too few astrometric points). When we add the uncertainties that are due to parallax (0.05 M$_\\odot$) and disk inclination (0.09 M$_\\odot$), the result is $1.85\\pm 0.27$~M$_\\odot$. \n\nThe mass estimated by this procedure might be underestimated because the photocenter of the blobs might be closer to the star than their center of mass and the mass determination depends on the cube of the separation. In particular, as noted above, we may interpret blob D as the accretion flows on a planet along a spiral arm (see the next section); in this case, the photocenter is dominated by the leading arm, which is at about 313 mas from the star, while the trailing arm is at 347 mas when it is deprojected on the disk plane. The putative planet would be in the middle of the two arms, that is, at 330 mas from the star, yielding a mass estimate that is $\\sim 10$\\% higher than listed in Table~\\ref{t:blobrot}. We note that the separation measured in the K2 band agrees very well with this interpretation.\n\nThe mass determined from blob motion is slightly higher than but in agreement within the error bars with the mass that fits photometry. To show this, we determined the stellar mass by minimizing the $\\chi^2$ with respect to the main-sequence values considered by \\cite{Pecaut2013}. We considered the GAIA DR2 parallax and included an absorption term $A_V$ multiplied for the reddening relation by \\cite{Cardelli1989}. We also left free the ratio between the stellar and the main-sequence radius. The best match is with an F0V star ($T_{\\rm eff}=7220$~K), with $A_V=0.25$~mag and a radius that is 0.97 times the radius of the main-sequence star. According to \\cite{Pecaut2013}, the mass of an F0V star is 1.59~M$_\\odot$. This spectral type compares quite well with the most recent determinations (A7V: \\cite{Dent2013}; A9V: \\cite{Vieira2003}; F0V: \\cite{Paunzen2001}; F1V: \\cite{Murphy2015}) and with the temperature determined by GAIA ($T_{\\rm eff}=7320\\pm 150$~K), but it is much later than the B9V spectra type proposed by \\cite{Wright2003}. \n\nFor comparison, other determinations of the mass of HD~169142 are 2.0~M$_\\odot$\\ (\\cite{Manoj2005}), 2.28~M$_\\odot$\\ (\\cite{Maaskant2013}), 1.8~M$_\\odot$\\ (\\cite{Salyk2013}), and 2.0~M$_\\odot$\\ (\\cite{Vioque2018}). The mass adopted by \\cite{Ligi2017} is 1.7~M$_\\odot$. We note that these values were obtained assuming distances different from the distance given by GAIA DR2: for instance, \\cite{Maaskant2013} adopted a distance of 145 pc, which is 27\\% larger than the GAIA DR2 value considered here. On the other hand, the value used by \\cite{Ligi2017} was taken from GAIA DR1 and it is only 3\\% longer than that from GAIA DR2.\nHereafter, we adopt a mass of 1.7~M$_\\odot$ for HD~169142, which is the same value as was considered by \\cite{Ligi2017}.\n\nWe also note that the projected rotational velocity of the star $V~\\sin{i}=50.3\\pm 0.8$~km\/s determined from the HARPS spectra (see Appendix) is high when we consider that the star is likely seen close to the pole. This value agrees quite well with literature values ($V~\\sin{i}=55\\pm 2$~km\/s: \\cite{Dunkin1997a, Dunkin1997b}). When we assume that the stellar rotation is aligned with the disk, the equatorial rotational velocity is 224~km\/s, which is at the upper edge of the distribution for F0 stars. For a discussion, see \\cite{Grady2007}.\n\n\n\\begin{table*}\n\\caption{Blob astrometry}\n\\begin{centering}\n\\begin{tabular}{lcccccccc}\n\\hline\n\\hline\nJD &\\multicolumn{2}{c}{Blob A}&\\multicolumn{2}{c}{Blob B}&\\multicolumn{2}{c}{Blob C}&\\multicolumn{2}{c}{Blob D}\\\\\n+2400000 & Sep & PA & Sep & PA & Sep & PA & Sep & PA \\\\\n & (mas) & (degree) & (mas) & (degree) & (mas) & (degree) & (mas) & (degree) \\\\\n\\hline\n57145. & $106\\pm 6$ & $247\\pm 3$ & $185.4\\pm 4.0$ & $22.3\\pm 1.0$ & $192.7\\pm 4.0$ & $315.8\\pm 2.0$ & $315.8\\pm 4.0$ & $43.8\\pm 0.7$ \\\\ \n57180.17 & & & $194.0\\pm 3.2$ & $24.8\\pm 1.0$ & $197.9\\pm 2.5$ & $316.0\\pm 0.7$ & $313.9\\pm 4.0$ & $40.3\\pm 0.7$ \\\\\n57201.12 & & & $188.3\\pm 3.2$ & $22.6\\pm 1.0$ & $202.8\\pm 2.5$ & $315.2\\pm 0.7$ & $314.8\\pm 4.0$ & $42.1\\pm 0.7$ \\\\\n57499.34 & & & $187.7\\pm 3.2$ & $21.0\\pm 1.0$ & $197.6\\pm 2.5$ & $313.2\\pm 0.7$ & & \\\\\n57566.15 & $125\\pm 6$ & $240\\pm 3$ & $188.7\\pm 3.2$ & $18.4\\pm 1.0$ & $203.7\\pm 2.5$ & $312.9\\pm 0.7$ & $315.5\\pm 4.0$ & $41.6\\pm 0.7$ \\\\\n57873.30 & $117\\pm 6$ & $230\\pm 3$ & $184.6\\pm 3.2$ & $14.0\\pm 1.0$ & $200.1\\pm 2.5$ & $307.5\\pm 0.7$ & $319.2\\pm 4.0$ & $40.2\\pm 0.7$ \\\\\n58288.19 & & & $189.7\\pm 3.2$ & $~9.8\\pm 1.0$ & $200.1\\pm 2.5$ & $299.7\\pm 0.7$ & $315.6\\pm 4.0$ & $34.9\\pm 0.7$ \\\\\n\\hline\n\\end{tabular}\n\\end{centering}\n\\label{t:blobastro}\n\\end{table*}\n\n\\begin{table*}\n\\caption{Blob rotation}\n\\begin{centering}\n\\begin{tabular}{lcccccccc}\n\\hline\n\\hline\nBlob & FAP & a & a & Period & Period & Rot. speed & Mass & Remark \\\\ \n & & & & Computed & Observed & & & \\\\\n & & (mas) & (au) & (yr) & (yr) & (deg\/yr) & (M$_\\odot$) & \\\\\n\\hline\n\nA & 0.02 & 118 & 13.5 & 36.2 & 42.7$\\pm$5.6 & -11.9$\\pm$2.2 \n& 1.60$\\pm$0.98 & pol and nocoro images \\\\\nB & $<1E-7$ & 188 & 21.4 & 72.5 & 78.3$\\pm$5.3 & -5.04$\\pm$0.38 & 2.30$\\pm$0.29 & \\\\\nC & $<1E-7$ & 202 & 23.1 & 80.9 & 73.0$\\pm$4.3 & -4.48$\\pm$0.25 & 1.60$\\pm$0.23 & \\\\\nD & 4E-6 & 319 & 36.4 & 160.5 & 173.8$\\pm$20.1 & -2.08$\\pm$0.25 & 1.34$\\pm$0.40 & \\\\\n\\hline\n\\end{tabular}\n\\\\\nNote: Semi-major axis a is obtained assuming circular orbits on the disk plane; the computed period is for a mass of 1.87~$M\\odot$; the observed period is estimated from the angular speed on the disk plane; the mass is determined using Kepler's third law; the uncertainty here is due to the errors in the angular speed.\n\\end{centering}\n\\label{t:blobrot}\n\\end{table*}\n\n\\subsection{Blob photometry}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{photo_blob.png}\n\\caption{Contrast of blobs as a function of wavelength. Blob B: Asterisks and dotted line. Blob C: Triangles and solid line. Blob D: Diamonds and dashed line.}\n\\label{f:contrast}\n\\end{figure}\n\nWe measured the magnitude of the sources in various bands by weighting the results obtained from the different epochs according to the quality of the images. The magnitudes refer to a $3\\times 3$ pixel area centered on each object and are obtained by comparison with those of simulated planets inserted into the image at 0.2 and 0.3 arcsec from the star and run through the same differential imaging algorithm. The underlying assumption is that the blobs are point sources, while they are likely slightly extended. These results should then be taken with caution. Using the fake blob procedure described in Section 3.2, we estimated that the brightness is underestimated by about a factor of $\\sim 2.8$~because of this effect for blobs B and C, that is, these blobs are likely $\\sim 1.1$ magnitude brighter than estimated when we assume that they are point sources. The effect is likely slightly smaller for blob D because it is farther away from the star. We summarize the results in Table~\\ref{t:blobphot}; error bars are the standard deviation of the mean of the results obtained at different epochs. All the blobs have a rather flat, only slightly reddish contrast with respect to the star (see Figure~\\ref{f:contrast}). Results are consistent with stellar light scattered by grains with a size on the order of a micron or smaller if stellar light is extinguished between the star and blobs or the blobs and us. Under the hypothesis (not demonstrated) that they are optically thick, the albedo required to reproduce observations of blobs A, B, and C is about 0.1.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{photo_blob_d.png}\n\\caption{Absolute magnitude of blob D in various bands (diamonds). The solid line is the prediction for a 3~M$_J$, 5~Myr old planet using dusty isochrones by \\cite{Allard2001}}\n\\label{f:photo_b}\n\\end{figure}\n\nBlob D is about two magnitudes fainter than expected from this consideration, suggesting that either it receives less light from the star (e.g., because of absorption by Ring 1) or it is not optically thick. For this blob, we obtained contrasts of $13.31\\pm 0.28$~mag in Y, $13.34\\pm 0.23$~mag in J band, $13.29\\pm 0.21$~mag in H band, $12.94\\pm 0.5$~mag in K1 band, and $12.35\\pm 0.5$~mag in K2 band (the last two values being obtained from the IRDIS data set). As expected, the object is beyond the 5$\\sigma$\\ contrast limit in each individual image. However, we expect a detection with an S\/N in the range from 2.3 to 3.7 in the individual images, and at an S\/N$\\sim 6$\\ in the combination of the images. It is then not surprising that we detected it only by combining them. While error bars are quite large, the absolute K1 and K2 magnitude of $14.07\\pm 0.50$~mag and $13.48\\pm 0.50$~mag corresponds to a $\\sim 3$~M$_J$ object using dusty isochrones (\\cite{Allard2001}) with an age of 5 Myr, which is at the lower edge of the age range according to \\cite{Blondel2006} and \\cite{Manoj2007}. This model has an effective temperature of about 1260 K (see Figure~\\ref{f:photo_b}). Of course, the mass estimated from photometry depends on the model, the age used to derive it and the possible extinction, and it assumes that the object is in hydrostatic equilibrium, which may be incorrect for a very young planet. This result is then highly uncertain.\n\n\\begin{table*}\n\\caption{Blob photometry. These values are obtained assuming that the blobs are point sources; they may be as much as 1.1 mag brighter if their extension is taken into account}\n\\begin{centering}\n\\begin{tabular}{lccccc}\n\\hline\n\\hline\nBlob & \\multicolumn{5}{c}{Contrast (in magnitudes) } \\\\\n & Y & J & H & K1 & K2 \\\\\n\\hline\nA & 9.05 & 8.73 & 9.02\\\\\nB & 10.06$\\pm$0.19 & 9.84$\\pm$0.01 & 9.67$\\pm$0.10 & 9.72$\\pm$0.21 & 9.54$\\pm$0.14 \\\\\nC & 10.43$\\pm$0.15 &10.26$\\pm$0.14 &10.09$\\pm$0.06 & 9.90$\\pm$0.40 & 9.86$\\pm$0.40 \\\\\nD & 13.31$\\pm$0.28 &13.34$\\pm$0.23 &13.29$\\pm$0.21 &12.94$\\pm$0.50 &12.35$\\pm$0.50 \\\\\n\\hline\n\\end{tabular}\n\\end{centering}\n\\label{t:blobphot}\n\\end{table*}\n\n\\subsection{Comparison with previous detection claims}\n\nWe note that none of these blobs coincides with either the sub-stellar companions proposed by \\cite{Biller2014} and \\cite{Reggiani2014}, nor with the structure observed by \\cite{Osorio2014}. More in detail, after taking into account their motion (see Section 3.3), the expected position angles for blobs A, B, and C at the observation epochs of Biller et al. and Reggiani et al. (both acquired at an epoch about 2013.5), are 276, 34, and 324 degree, respectively (blob D is much farther away from the star). For comparison, the object of Biller et al. is at PA=$0\\pm 14$~degree (separation of $110\\pm 30$\\ mas) and the object of Reggiani et al. is at PA=$7.4\\pm 11.3$ degree (separation of $156\\pm 32$~mas). In addition, the objects proposed by Biller et al. and Reggiani et al., with a contrast of $\\Delta L\\sim 6.5$, are brighter than our blobs B and C, even after the finite-size correction is taken into account, see Section 3.4. However, the object proposed by Reggiani et al. might be the combination of blobs B and C, within the errors of their astrometry; the combination of their luminosity is also not that far from the value of Reggiani et al. We note that the resolution of their observation is lower than ours because they observed at much longer wavelength, and their object appears elongated (in the E-W direction, i.e., the direction expected at the epoch of their observation) in their published image beyond the diffraction limit.\n\nOn the other hand, the inner and brighter object detected by Biller et al. is too close to the star to coincide with any of the objects we observed, while a fainter object they found might be blob B, as discussed by \\cite{Ligi2017}. However, when we examine the image published by Biller et al, it seems that the two brightest sources have a relative separation and orientation that coincides with those of blobs B and C. In this case, the fainter object should be blob B (as discussed in \\cite{Ligi2017}) and the brighter object might coincide with our blob C. Of course, this would require that the stellar position in their images does not correspond with the position assumed in their paper.\n\n\n\\section{Spiral arms}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=8.8truecm]{polar.jpeg}\n\\caption{Median over time of the individual S\/N maps in polar coordinates. Each image has been rotated to the last image for the rotation angle of blob D before the median was made. Arrows mark the location of the primary (white), secondary (cyan), and tertiary arms (yellow). The location of the blobs is marked.}\n\\label{f:polar}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{nospiral.jpeg}\n\\includegraphics[width=\\columnwidth]{spiral.jpeg}\n \n \n\\caption{Upper panel: Median over time of the individual S\/N maps; each image has been rotated to the last image for the rotation angle of blob D before the median was made. Lower panel: Same as the upper panel, with a three-arm spiral design overplotted. The putative planet is at the position of blob D. In both panels, N is up and E to the left, and a white cross marks the stellar position.}\n\\label{f:spirals}\n\\end{figure}\n\nMost bright features in the coincidence images (including blobs B and D) can be reproduced by a three-arm spiral design (see below), while blob C differs slightly. A similar three-arm structure is predicted by models for not very massive companions (\\cite{Fung2015}) and it has been observed around other stars (see, e.g., the case of MWC758 recently published by \\cite{Reggiani2018}). While it is not at all obvious that spiral arms indicate a planet (see, e.g., \\cite{Dong2018}), we may interpret it as due to a planet in the location of blob D. The structure of this blob appears to resemble the structure expected for an accreting object with a leading and a trailing arm. If this hypothesis were correct, the radial separation between leading (sep=310 mas) and trailing (sep=343 mas) arms should be about twice the Hill radius (see \\cite{Machida2010}); the Hill radius would then be $16.5\\pm 5$~mas and the planet mass in the range 0.25-1.6~M$_J$, which is lower than the mass estimated with DUSTY isochrones. Because the Hill radius is not accurately estimated and the dependence of the mass on the Hill radius is strong, the error on the planetary mass is quite large. The photosphere of such an object would be too faint for detection in YJH, while it might be compatible with detection in K1 and K2 bands.\n\n\\subsection{Separation of spiral arms}\n\nWe may also estimate the mass of the object exciting the spiral design observed in Ring 1 by different criteria, using the calibration by \\cite{Fung2015} (their eq. (9)). After transforming into a polar coordinates system (see Figure~\\ref{f:polar}), we could identify the three spiral arms, which we may call primary, secondary, and tertiary, following the approach of Fung \\& Dong. The view in Cartesian coordinates is given in Figure~\\ref{f:spirals}. Position angle and separation of the arms in some reference positions are given in Table~\\ref{t:spiral}.\n\n\\begin{table}\n\\caption{Spiral position.}\n\\begin{tabular}{lccc}\n\\hline\n\\hline\n & PA & PA & PA \\\\\nSep & Primary & Secondary & Tertiary \\\\\n(mas) & (degree) & (degree) & (degree) \\\\\n\\hline\n157 & 196.7 & 321.0 & 38.9 \\\\\n172 & 203.0 & 335.9 & 50.9 \\\\\n194 & 244.2 & 16.0 & 79.0 \\\\\n209 & 268.3 & 28.0 & 91.6 \\\\\n\\hline\npitch & 15.3 & 16.3 & 20.8 \\\\\n\\hline\n\\end{tabular}\n\\label{t:spiral}\n\\end{table}\n\nThese arms may be density waves excited by a a planet at the location of blob D, which is indeed along the primary arm of the spiral design: the predicted PA at the separation of blob D, 334 mas, is $36\\pm 6$~degree, in very good agreement with the observed value of $\\sim 35$~degree (as measured in the K2 band).\n\nThe phase difference between the primary and secondary arm ($127.2\\pm 3.1$~degree) can be used to estimate the mass of the planet exciting the spiral design, using the calibration by Fung \\& Dong. We obtain a mass ratio of $q=0.0030\\pm 0.0004$, which translates into a mass of $M_p=5.1\\pm 1.1$~M$_J$, adopting the stellar mass derived above. The phase difference between secondary and tertiary arms ($69.9\\pm 3.9$~degree) agrees with the expectations by Fung \\& Dong given the pitch angle and the expected ratio for resonances 1:2 and 1:3.\n\n\\subsection{Pitch angle}\n\nThe pitch angle is the angle between a spiral arm and the tangent to a circle at the same distance from the star. \\cite{Zhu2015} showed that the pitch angle can be used to estimate the mass of the planet exciting the spiral design. We measured the pitch angle at a separation of 183 mas to be $17.5\\pm 1.7$~degree. This separation is about $r\/r_p=0.55$. This value agrees with the results they obtained from their simulations for a mass ratio of $q=0.006$, supporting the mass determination obtained from the separation of primary and secondary spiral arm; moreover, the larger pitch for the tertiary arm agrees with expectations from models.\n\n\\subsection{Disk gap}\n\nUsing the relation by \\cite{Kanagawa2016}, we expect that there is a planet at $\\sim 0.36$~arcsec from the star with a mass ratio with respect to the star of $q=2.1\\times 10^{-3}\\,(W\/R_p)^2\\,(h_p\/0.05~R_p)^{1.5}\\,(\\alpha\/10^{-3})^{0.5}$, where $h_p\/R_p$ is the disk thickness and $\\alpha$\\ is the disk viscosity. For $R_p=0.36$~arcsec, $W=0.2$~arcsec, $h_p\/R_p=0.05$, and $\\alpha=1E-3$, a value of $q=0.00044$\\ is obtained, which means a planet of 0.75~M$_J$. \n\n\\cite{Dong2017} considered the case of HD~169142 and concluded for a value of $q^2\/\\alpha=1.1E-4$\\ for $R_p=0.37$~arcsec, $W=0.17$\\ arcsec, and $h_p\/R_p$=0.079. For $\\alpha=1E-3$, their formula implies $q=0.00033,$\\ which suggests a 0.56~M$_J$\\ planet. We note that the formula by Dong \\& Fung produces planets that are smaller by a factor of 2.6 with respect to that by Kanagawa et al.; however, the value they suggest for $h_p\/R_p$\\ is higher than considered above. The value considered by Dong \\& Fung is similar to the value obtained by \\cite{Fedele2017} by modeling the ALMA observations ($h_p\/R_p$=0.07).\n\nThere are considerable uncertainties in these formulas that are due to the exact values to be adopted for $R_p$, $W$, $h_p\/R_p$, $\\alpha$, and the difference of a factor of 2.5 in the constant factor. While a mass around 1~M$_J$\\ seems favored, we cannot exclude values different by as much as an order of magnitude. We conclude that a planet with about one Jupiter mass likely causes the gap seen in HD169142, but its mass is not yet well defined from the gap alone.\n\n\\begin{table}\n\\caption{Putative planet mass (sep=335 mas, PA=35 degree at JD=58288.19)}\n\\begin{tabular}{lcc}\n\\hline\n\\hline\nMethod & M$_J$ & Remark \\\\\n\\hline\nPhotometry & 3 & Age dependent \\\\\nHill radius & 0.25-1.6 & \\\\\nSpiral arm separation & 4.0-6.2 & \\\\\nPitch angle & 6 & \\\\\nDisk gap & 0.06-6 & \\\\\n\\hline\n\\end{tabular}\n\\label{t:mass}\n\\end{table}\n\n\\subsection{Summary of mass determination}\n\nA summary of the mass determinations is given in Table~\\ref{t:mass}. All these estimates are quite uncertain. The higher values are given by the spiral arm parameters. If we make an harmonic mean of the various estimates, we would conclude for a planet with a mass of $2.2_{-0.9}^{+1.4}$~M$_J$. This mass seems lower than what we can detect with our SPHERE images (about 3~M$_J$\\ from photometry), but is within the error bar. This value is also within the range 1-10~M$_J$\\ suggested by \\cite{Fedele2017} to justify the dust cavity observed with ALMA between rings 1 and 2, and it is on the same order as the missing mass in the disk within the gap, as given by their disk model (4.3~M$_J$).\nFor comparison, we note that if we were to try to interpret the spiral arms of MWC~758 (\\cite{Reggiani2018}) using the same approach, we would conclude for a more massive faint companion because in that case the separation between the primary and secondary arm is much closer to 180 degree.\n \n\n\n\n\n\n\n\n\\section{Conclusion}\n\nWe performed an analysis of faint structures around HD~169142 that are persistent among several data sets obtained with SPHERE and analyzed them using differential image techniques. We found a number of blobs that rotate around the star as well as spiral arms. These structures represent small fluctuations of the overall disk structure around this star. The blobs are found to consistently rotate around the star with Keplerian circular motion.\n\nAlthough we cannot exclude other hypotheses, blob D might correspond to a low-mass ($\\sim 1-4$~M$_J$, best guess of 2.2~M$_J$), 5 Myr old, and still-accreting planet at about 335 mas (38 au) from the star, causing the gap between Rings 1 and 2 and exciting the spiral arm design observed within Ring 1. The separation between the outer edge of Ring 1 and blob D is 55 mas, which is about twice the proposed value for the Hill radius of the planet. There is a clear excess of flux at short wavelengths with respect to the flux expected for a planetary photosphere (see Figure~\\ref{f:photo_b}). In our proposed scenario, the planetary photosphere is not detected in YJH band, where we only see the accreting material fueling through the spiral arm and reflecting star light (consistent with its detection in the $Q_\\Phi$\\ image), while it might have been detected in the K1 and K2 bands. A planet of 2.2~$M_J$\\ at 335~mas (38~au) from HD~169142 would have a Hill radius of about 25 mas (3.2 au). A disk around such an object would have an FWHM slightly larger than the resolution limit of SPHERE and may well reflect some $10^{-5}$ of the stellar light, which is required to justify the flux observed in the YJH bands. On the other hand, it is also possible that no other planet exists, and we merely observe a dust cloud. Detection of a planet could be confirmed by observations in the L' band. According to the AMES-dusty isochrones (\\cite{Allard2001}), a 5 Myr old planet of 2.2~$M_J$\\ should have an absolute L' magnitude of $\\sim 11$~mag. The contrast in the L' band should then be of 10.1 mag, which is 3.7 mag fainter than the objects proposed by \\cite{Biller2014} and \\cite{Reggiani2014} and likely too faint for a detection in their data set. However, a future deeper data set can solve this issue.\n\nThe location of blob B (and C to a lesser extent) suggests at first sight that the blobs might be related to the secondary and tertiary spiral arms (see, e.g., \\cite{Crida2017}). If this were the case, they would follow the same angular speed as the perturbing object, that is, the putative planet. However, we showed (along with Ligi et al.) that those blobs follow a Keplerian motion appropriate for their separation from the star.\n\nFinally, we note that \\cite{Ligi2017} proposed that blobs B and C could be vortices (\\cite{Meheut2012}). This explanation might very well be true. Another scenario might be suggested by the possibility that they are in 1:2 resonance with a putative planet related to blob D. It concerns planetesimals or asteroid giant impacts that generate dust clouds. This might be a manifestation of the general phenomenon of planetesimal erosion that is expected to follow the formation of giant planets (see, e.g., \\cite{Turrini2012, Turrini2018}). However, the probability of observing such clouds is low in a gas-rich disk such as that of HD~169142 because large planetesimals are required to generate clouds as large as blobs B and C, unless the impact occurs far from the disk plane. The debris cloud from an impact roughly expands until the debris sweeps a gas mass that is no more than an order of magnitude higher than the mass of the debris itself. Because the volume of the clouds is at least one hundredth of the total volume of ring 1, this requires that the mass of the interacting bodies is higher than 1\/1000 of the mass of the disk when we assume a disk gas-to-dust ratio of unity and that the impact occurs close to the disk plane. The impacting bodies should then have a mass on the order of that of Mars or at least the Moon. Since it is not likely that many such objects are present in the disk of HD~169142, the probability of observing one or even more similar debris clouds is likely very low.\n\n\n\n\\begin{acknowledgements}\nThe authors thank A. Pohl for allowing them to use the original reduction of the $Q_\\Phi$ data set and the ESO Paranal Staff for support for conducting the observations. E.S., R.G., D.M., S.D. and R.U.C. acknowledge support from the \"Progetti Premiali\" funding scheme of the Italian Ministry of Education, University, and Research. E.R. and R.L. are supported by the European Union's Horizon 2020 research and innovation programme under the Marie Sk\u0142odowska-Curie grant agreement No 664931. This work has been supported by the project PRININAF 2016 The Cradle of Life - GENESIS-SKA (General Conditions in Early Planetary Systems for the rise of life with SKA). The authors acknowledge financial support from the Programme National de Plan\\'etologie (PNP) and the Programme National de Physique Stellaire (PNPS) of CNRS-INSU. This work has also been supported by a grant from the French Labex OSUG\\@2020 (Investissements d'avenir - ANR10 LABX56). The project is supported by CNRS, by the Agence Nationale de la Recherche (ANR-14-CE33-0018). This work is partly based on data products produced at the SPHERE Data Centre hosted at OSUG\/IPAG, Grenoble. We thank P. Delorme and E. Lagadec (SPHERE Data Centre) for their efficient help during the data reduction process. SPHERE is an instrument designed and built by a consortium consisting of IPAG (Grenoble, France), MPIA (Heidelberg, Germany), LAM (Marseille, France), LESIA (Paris, France), Laboratoire Lagrange (Nice, France), INAF Osservatorio Astronomico di Padova (Italy), Observatoire de Gen\u00e8ve (Switzerland), ETH Zurich (Switzerland), NOVA (Netherlands), ONERA (France) and ASTRON (Netherlands) in collaboration with ESO. SPHERE was funded by ESO, with additional contributions from CNRS (France), MPIA (Germany), INAF (Italy), FINES (Switzerland) and NOVA (Netherlands). SPHERE also received funding from the European Commission Sixth and Seventh Framework Programmes as part of the Optical Infrared Coordination Network for Astronomy (OPTICON) under grant number RII3-Ct-2004-001566 for FP6 (2004-2008), grant number 226604 for FP7 (2009-2012), and grant number 312430 for FP7 (2013-2016).\n\\end{acknowledgements}\n\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{#1}\n\\setcounter{equation}{0}}\n\\newcommand{\\mysubsection}[1]{\\subsection{#1}\n\\setcounter{equation}{0}}\n\n\n\\newcommand{\\ensuremath{X}}{\\ensuremath{X}}\n\n\\renewcommand{\\text{Im}}{\\ensuremath{{\\operatorname{Im}}}}\n\\renewcommand{\\Re}{\\ensuremath{{\\operatorname{Re}}}}\n\\newcommand{\\ensuremath{\\mathds 1}}{\\ensuremath{\\mathds 1}}\n\\newcommand{\\feinschr}[2]{#1\\big|_{#2}}\n\n\\newcommand{\\ensuremath{\\mathbb P}}{\\ensuremath{\\mathbb P}}\n\\newcommand{\\ensuremath{\\mathds E}}{\\ensuremath{\\mathds 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\\textbf{#1.}]\\ignorespaces\n}\n\n\n\\excludecomment{suggestion}\n\n\\excludecomment{pcidetail}\n\n\n\\definecolor{felix}{rgb}{0.2,0.2,1.0}\n\\definecolor{petru}{rgb}{0.7,0.1,0.1}\n\\definecolor{alternative}{rgb}{0.1,0.1,0.7}\n\\definecolor{detail}{rgb}{0.0,0.5,0.0}\n\n\\newcommand{\\color{black}}{\\color{black}}\n\\newcommand{\\color{black}}{\\color{black}}\n\n\\newcommand{\\color{black}}{\\color{black}}\n\\newcommand{\\color{black}}{\\color{black}}\n\n\\newcommand{\\color{black}}{\\color{felix}}\n\\newcommand{\\color{black}}{\\color{black}}\n\n\\newcommand{\\color{black}}{\\color{petru}}\n\\newcommand{\\color{black}}{\\color{black}}\n\n\\newcommand{\\color{alternative}}{\\color{alternative}}\n\\newcommand{\\color{black}}{\\color{black}}\n\n\\newcommand{\\color{detail}}{\\color{detail}}\n\\newcommand{\\color{black}}{\\color{black}}\n\n\n\\begin{document}\n\n\\title[The stochastic heat equation on polygonal domains]\n{On the regularity of the stochastic heat equation\n on polygonal domains in $\\bR^2$}\n\n\n\n\\author{Petru A. Cioica-Licht}\n\\thanks{The first named author has been partially supported by the Marsden Fund Council from Government funding, administered by the Royal Society of New Zealand, and by a University of Otago Research Grant (114023.01.R.FO). The research of the second and third author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) \nfunded by the Ministry of Education (NRF-2017R1D1A1B03033255) and (NRF-2013R1A1A2060996), respectively.\nThe authors would like to thank Felix Lindner for his contribution at an early stage of this manuscript}\n\\address{Petru A. Cioica-Licht (n\\'e Cioica), Department of Mathematics and Statistics, University of Otago, PO Box~56, Dunedin 9054, New Zealand}\n\\email{pcioica@maths.otago.ac.nz}\n\\author{Kyeong-Hun Kim}\n\\address{Kyeong-Hun Kim, Department of Mathematics, Korea University, Anam-ro 145, Sungbuk-gu, Seoul, 02841, Republic of Korea}\n\\email{kyeonghun@korea.ac.kr}\n\\author{Kijung Lee}\n\\address{Kijung Lee, Department of Mathematics, Ajou University, Worldcup-ro 206, Yeongtong-gu, Suwon, 16499, Republic of Korea}\n\\email{kijung@ajou.ac.kr}\n\n\n\\subjclass[2010]{60H15; 35R60, 35K05}\n\n\\keywords{Stochastic partial differential equation,\nstochastic heat equation,\nweighted $L_p$-estimate,\nweighted Sobolev regularity,\nangular domain,\npolygon,\npolygonal domain,\nnon-smooth domain,\ncorner singularity}\n\n\\begin{abstract}\nWe establish existence, uniqueness and higher order weighted $L_p$-Sobolev regularity for the stochastic heat equation with zero Dirichlet boundary condition on angular domains and on polygonal domains in $\\mathbb{R}^2$.\nWe use a system of mixed weights consisting of appropriate powers of the distance to the vertexes and of the distance to the boundary to measure the regularity with respect to the space variable.\nIn this way we can capture the influence of both main sources for singularities: the incompatibility between noise and boundary condition on the one hand and the singularities of the boundary on the other hand.\nThe range of admissible powers of the distance to the vertexes is described in terms of the maximal interior angle and is sharp. \n\\end{abstract}\n\n\\maketitle\n\n\n\\mysection{Introduction}\\label{sec:Introduction}\n\nIn this article we continue the analysis started in~\\cite{CioKimLee+2018} towards a refined $L_p$-theory for second order stochastic partial differential equations (SPDEs, for short) on non-smooth domains.\nThe main challenges in the construction of such a theory come from two effects that are known to influence the regularity of the solution:\nOn the one hand, the incompatibility between noise and boundary condition results in blow-ups of the higher order derivatives near the boundary---even if the boundary is smooth. \nOn the other hand, the singularities of the boundary cause a similar effect in their vicinity---even if the forcing terms are deterministic.\nWe refer to the introduction of~\\cite{CioKimLee+2018} and the literature therein for details. \n\nThe well developed $L_p$-theory for second order SPDEs on smooth domains, \ncarried out within the analytic approach initiated by N.V.~Krylov,\nshows that the incompatibility between noise and boundary condition can be captured very accurately by using a system of weights based on the distance to the boundary, see, for instance,~\\cite{Kim2004, KimKry2004, Kry1994,KryLot1999, KryLot1999b}.\nMoreover, the results in~\\cite{CioKimLee+2018} indicate that a system of weights based on the distance to a corner of the underlying domain is suitable to describe the impact of this boundary singularity on the solution.\nThus, in order to capture both effects, a system based on a combination of appropriate powers of the distance to the boundary and of the distance to the boundary singularities suggests itself.\n\n\nOur primary goal in this article is to show how such a system of mixed weights can be used in order to provide higher order spatial weighted $L_p$-Sobolev regularity for second order SPDEs with zero Dirichlet boundary condition on angular domains and on polygonal domains in $\\bR^2$. \nFor the moment we restrict ourselves to the stochastic heat equation, since already the analysis of this equation involves many non-trivial steps and has been a persisting problem for a long time.\nAt the same time, we believe that in this way we can shade some light on the general strategy without getting lost in details.\nOur general setting is as follows: \nLet $(w^k_t)$, $k\\in\\bN$, be a sequence of independent real-valued standard Brownian motions on a probability space $(\\Omega,\\cF,\\ensuremath{\\mathbb P})$ and let $T\\in (0,\\infty)$ be a finite time horizon.\nWe consider the stochastic heat equation \n\\begin{equation}\\label{eq:SHE:Intro:a}\n\\left.\n\\begin{alignedat}{3}\ndu \n&= \n\\grklam{ \\Delta && u + f^0+f^i_{x_i}}\\,dt\n+\ng^k\\,dw^k_t \\quad \\text{on } \\Omega\\times(0,T]\\times\\domain,\t\\\\\nu\n&=\n0 && \\quad \\text{on } \\Omega\\times(0,T]\\times\\partial\\domain,\t\\\\\nu(0)\n&=\n0 && \\quad \\text{on } \\Omega\\times\\domain,\n\\end{alignedat}\n\\right\\}\t\n\\end{equation}\non various types of domains $\\domain\\subseteq\\bR^d$.\nOur focus lies in particular on polygonal domains and on angular domains $\\domain \\subseteq\\bR^2$. \nNote that, as usual, here and in the sequel we use the Einstein summation convention on the repeated indexes $i$ and $k$.\n\n\nOur main results address the existence, uniqueness and higher order spatial regularity of the solution to Equation~\\eqref{eq:SHE:Intro:a} on angular domains and on polygonal domains $\\domain\\subset\\bR^2$. \nBy using a weight system based solely on the distance to the set of vertexes of $\\domain$, we establish existence and uniqueness of a solution to Equation~\\eqref{eq:SHE:Intro:a} with suitable weighted $L_p$-Sobolev regularity of order one with respect to the space variable; see Theorem~\\ref{thm:ex:uni:2DCone} (angular domains) and Theorem~\\ref{thm polygon main} (polygonal domains). \nThe lower bound of the range~\\eqref{eq:range:vertex} for the weight parameter $\\theta$, which corresponds to the best integrability property of the solution near the vertex, is sharp; see also the introduction of~\\cite{CioKimLee+2018}. \nMoreover, by using, in addition, appropriate powers of the distance to the boundary $\\partial \\domain$ we describe the behavior of higher order spatial derivatives of the solution; see Corollary~\\ref{high} (angular domains) and Theorem~\\ref{thm_polygons_1} (polygonal domains). \n\nThe key estimate, which paves the way for all the results mentioned above, is presented in Theorem~\\ref{lem:estim:2DCone:Theta}. \nRoughly speaking, it shows which system of weights is suitable in order to be able to lift the spatial regularity of the solution of the stochastic heat equation~\\eqref{eq:SHE:Intro:a} on an angular domain with the regularity of the forcing terms. In short, it can be stated as follows:\nLet \n\\begin{equation}\\label{domain:angular}\n\\cD:=\\cD_{\\kappa_0}:=\\ggklam{x\\in \\bR^2: x=(r\\cos\\vartheta,r\\sin\\vartheta),\\; r>0,\\;\\vartheta\\in (0,\\kappa_0)},\n\\end{equation}\nbe an angular domain with vertex at the origin and angle~$\\kappa_0\\in(0,2\\pi)$.\nMoreover, let $\\dist(x):=\\dist_\\cD(x):=\\mathrm{dist}(x,\\partial\\cD)$ be the distance of a point $x\\in\\cD$ to the boundary $\\partial\\cD$ of $\\cD$. If $u$ is the solution to Equation~\\eqref{eq:SHE:Intro:a} on $\\cD$,\nthen, for arbitrary $m\\in \\bN$, $1<\\Theta0$ that depends on the roughness of the boundary of the domain and is not explicitly given~\\cite{Kim2014}. In particular, for large $p>2$, $\\Theta=d$ is not admissible, see~\\cite[Example~2.17]{Kim2014} for a typical counterexample.\nOur results show that, on polygonal domains, if we use an appropriate power of the distance to the set of vertexes to control the behavior of the solution in their proximate vicinity, then $\\Theta=d=2$ is possible away from the vertexes.\n\nOur analysis takes place within the framework of the analytic approach.\nThe proofs of the main results rely on a mixture of Green function estimates on angular domains, suitable localization techniques and some delicate estimates for the stochastic heat equation on $\\ensuremath{\\mathcal{C}}^1$ domains.\nAlternatively, one could think of Equation~\\eqref{eq:SHE:Intro:a} as an abstract Banach space valued stochastic evolution equation and try to obtain a similar theory by using the extension of the semigroup approach for SPDEs to Banach spaces developed by J.M.A.M.~van Neerven, M.C.~Veraar and L.~Weis~\\cite{NeeVerWei2008, NeeVerWei2012, NeeVerWei2012b}. However, for this to succeed, one would have to (at least!) check whether the (properly defined) Dirichlet Laplacian on weighted Sobolev spaces has an appropriate functional calculus. \nMoreover, one would need a description of the domain of the square root of this operator in terms of suitable weighted Sobolev spaces.\nTo the best of our knowledge, both questions are not trivial and yet to be answered.\nIn this context it is worth mentioning that the recently developed Calder\\'on-Zygmund theory for singular stochastic integrals from~\\cite{LorVer2019+} together with the $L_p$-theory developed in~\\cite{CioKimLee+2018} lead to an $L_q(L_p)$-theory with $q\\neq p$ without making use of precise descriptions of the domains of fractional powers of the Laplacian nor of the existence of a bounded $H^\\infty$-calculus, see~\\cite[Example~8.12]{LorVer2019+}.\n\nThis article is organized as follows: In Section~\\ref{sec:2DCone} we present and prove the main results concerning existence, uniqueness (Theorem~\\ref{thm:ex:uni:2DCone}) and higher order regularity (Corollary~\\ref{high}) of the stochastic heat equation on angular domains.\nThe proofs rely on two key estimates, which are stated in Theorem~\\ref{lem:estim:2DCone:Theta} and Lemma~\\ref{lem 4.5.1} and proven in detail in Section~\\ref{sec:proof:lift} and Section~\\ref{4}, respectively. \nFinally, in Section~\\ref{sec:Polygons} we present our analysis of the stochastic heat equation on polygonal domains.\nBefore we start, we fix some notation.\n\n\\medskip\n\n\\noindent\\textbf{Notation.} \nThroughout this article, $(\\Omega,\\cF,\\ensuremath{\\mathbb P})$ is a complete probability space and $\\nrklam{\\cF_{t}}_{t\\geq0}$ is an increasing filtration of $\\sigma$-fields $\\cF_{t}\\subset\\cF$, each of which contains all $(\\cF,\\ensuremath{\\mathbb P})$-null sets.\nWe assume that on $\\Omega$ we are given a family $(w_t^k)_{t\\geq0}$, $k\\in\\bN$, of independent one-dimensional Wiener processes relative to $\\nrklam{\\cF_{t}}_{t\\geq0}$. By $\\cP$ we denote the predictable $\\sigma$-algebra on $\\Omega\\times (0,\\infty)$ generated by $\\nrklam{\\cF_{t}}_{t\\geq0}$ and any of its trace $\\sigma$-algebras.\nMoreover, $T\\in(0,\\infty)$ is a finite time horizon and $\\Omega_T:=\\Omega\\times (0,T]$. \nFor a measure space $(A, \\cA, \\mu)$, a Banach space $B$ and $p\\in[1,\\infty)$, we write $L_p(A,\\cA, \\mu;B)$ for the collection of all $B$-valued $\\bar{\\cA}$-measurable functions $f$ such that \n$$\n\\|f\\|^p_{L_p(A,\\cA,\\mu;B)}:=\\int_{A} \\lVert f\\rVert^p_{B} \\,d\\mu<\\infty.\n$$\nHere $\\bar{\\cA}$ is the completion of $\\cA$ with respect to $\\mu$. The Borel $\\sigma$-algebra on a topological space $E$ is denoted by $\\cB(E)$. We will drop $\\cA$ or $\\mu$ in $L_p(A,\\cA, \\mu;B)$ when the $\\sigma$-algebra $\\cA$ or the measure $\\mu$ are obvious from the context. \nFor functions $f$ depending on $\\omega\\in \\Omega$, $t\\geq 0$ and $x\\in\\bR^d$, we usually drop the argument $\\omega$, and denote them by $f(t,x)$. \nIf $\\domain\\subseteq\\bR^d$ is a domain in $\\bR^d$, we write $\\ensuremath{\\mathcal{C}}^{\\infty}_c(\\domain)$ for the space of infinitely differentiable functions with compact support in $\\domain$. Moreover, $\\ensuremath{\\mathcal{C}}^{2}_c(\\domain)$ is the space of twice continuously differentiable functions with compact support in $\\domain$.\nFor a function $f\\colon\\domain\\to\\bR$ and any multi-index $\\alpha=(\\alpha_1,\\ldots,\\alpha_d)$, $\\alpha_i\\in \\{0,1,2,\\ldots\\}$, \n$$\nD^{\\alpha}f(x):=\\partial^{\\alpha_d}_d\\cdots\\partial^{\\alpha_1}_1u(x),\n\\quad x=(x^1,\\ldots,x^d),\n$$\nwhere $\\partial^{\\alpha_i}_i=\\frac{\\partial^{\\alpha_i}}{\\partial (x^i)^{\\alpha_i}}$ is the $\\alpha_i$ times (generalized) derivative with respect to the $i$-th coordinate;\n$f_{x^i}:=\\frac{\\partial}{\\partial x^i}u$.\nBy making slight abuse of notation, for $m\\in\\{0,1,2,\\ldots\\}$, we write $D^m f$ for any (generalized) $m$-th order derivative of $f$ and for the vector of all $m$-th order derivatives. For instance, if we write $D^mf\\in B$, where $B$ is a function space on $\\domain$, we mean $D^\\alpha\\in B$ for all multi-indexes $\\alpha$ with $\\abs{\\alpha}=m$.\nThe notation $f_x$ is used synonymously for $D^1f$, whereas $\\nnrm{f_x}{B}:=\\sum_i\\nnrm{f_{x^i}}{B}$.\nThroughout the article, the letter $N$ is used to denote a finite positive constant that may differ from one appearance to another, even in the same chain of inequalities.\nWhen we write $N=N(a,b,\\cdots)$, we mean that $N$ depends only on the parameters inside the parentheses.\n Moreover, $A\\sim B$ is short for `$A\\leq N B$ and $B\\leq N A$'. \n\n\n\n\n\n\\mysection{The stochastic heat equation on angular domains }\\label{sec:2DCone}\n\n\n\nIn this section we present our analysis for the stochastic heat equation \n\\begin{eqnarray}\\label{eq:SHE:Intro}\ndu=(\\Delta u+f^0+ f^i_{x^i})\\,dt+ g^k \\,dw^k_t, \\quad t\\in (0,T],\n\\end{eqnarray}\non angular domains $\\cD\\subseteq\\bR^2$ with zero Dirichlet boundary condition and vanishing initial value.\nWe establish existence and uniqueness (Theorem~\\ref{thm:ex:uni:2DCone}) as well as higher order spatial regularity of the solution (Corollary~\\ref{high}) within a framework of weighted Sobolev spaces. \nThe weights are products of appropriate powers of the distance to the vertex and of the distance to the boundary (two infinite edges and the vertex). \nThe key estimate, which enables us to describe the behavior of the higher order derivatives of $u$ near the boundary even if the forcing terms behave badly near the boundary but are sufficiently smooth inside the domain, is presented in Theorem~\\ref{lem:estim:2DCone:Theta}, see also Remark~\\ref{explanation of key}.\n\n\n\n To state our results, we first introduce appropriate function spaces.\nThe notation is mainly borrowed from~\\cite{CioKimLee+2018}. Throughout, $\\cD=\\cD_{\\kappa_0}$ is as defined in~\\eqref{domain:angular} with $\\kappa_0\\in(0,2\\pi)$ and $\\rho_\\circ(x):=\\abs{x}$ denotes the distance of a point $x\\in \\cD$ to the origin (the only vertex of $\\cD$).\nLet $p>1$ and $\\theta\\in\\bR$.\nWe write\n$$\nL^{[\\circ]}_{p,\\theta}(\\cD):=L_{p}(\\cD,\\cB(\\cD),\\rho_\\circ^{\\theta-2} dx;{\\mathbb R}}\\def\\RR {{\\mathbb R})\n\\quad\\text{and}\\quad\nL^{[\\circ]}_{p,\\theta}(\\cD; \\ell_2):=L_p(\\cD,\\cB(\\cD),\\rho_\\circ^{\\theta-2} dx;\\ell_2)\n$$\nfor the weighted $L_p$-spaces of real-valued and $\\ell_2$-valued functions with weight $\\rho_\\circ^{\\theta-2}$. For $n\\in\\{0,1,2,\\ldots\\}$ let\n\\[\n\\ensuremath{K}^n_{p,\\theta}(\\cD)\n=\n\\ssggklam{\nf : \\nnrm{f}{\\ensuremath{K}^n_{p,\\theta}(\\cD)} := \\ssgrklam{\\sum_{\\abs{\\alpha}\\leq n} \n\\gnnrm{\\rho_{\\circ}^{\\abs{\\alpha}} D^\\alpha f}{L_{p,\\theta}^{[\\circ]}(\\cD)}^p}^{1\/p}\n< \n\\infty\n},\n\\]\nand define $K^n_{p,\\theta}(\\cD;\\ell_2)$ accordingly. Note that\n\\[\nK^0_{p,\\theta}(\\cD)=L^{[\\circ]}_{p,\\theta}(\\cD)\n\\quad\\text{and}\\quad\nK^0_{p,\\theta}(\\cD;\\ell_2)=L^{[\\circ]}_{p,\\theta}(\\cD;\\ell_2).\n\\]\nMoreover, we write $\\mathring{\\ensuremath{K}}^1_{p,\\theta}(\\cD)$ for the closure in $\\ensuremath{K}^1_{p,\\theta}(\\cD)$ of the space $\\ensuremath{\\mathcal{C}}^\\infty_c(\\cD)$ of test functions.\n\nThe weighted Sobolev spaces introduced above are classical examples of Kondratiev spaces. For their basic properties as well as their relevance in the analysis of elliptic partial differential equations on domains with conical singularities we refer to~\\cite[Part~2]{KozMazRos1997}, see also the pioneering works \\cite{Kon1967,Kon1970,KonOle1983, KufOpi1984}. \nIn the sequel, we will frequently use the following basic properties.\nThey are mainly a consequence of the fact that \nfor any multi-index $\\alpha$\n$$\n\\sup_{\\cD} \\left(\\rho^{|\\alpha|-1}_{\\circ} |D^{\\alpha}\\rho_{\\circ}|\\right)\\le N(\\alpha)<\\infty;\n$$\nthe proof is left to the reader. \n\\begin{lemma}\n\\label{lem 1}\nLet $p>1, \\theta\\in \\bR$ and $n\\geq 1$. If $\\alpha$ is a multi-index with \n$|\\alpha|\\leq n$, then\n$$\n\\|\\rho_{\\circ}^{|\\alpha|} D^{\\alpha} f\\|_{K^{n-|\\alpha|}_{p,\\theta}(\\cD)}+\\|D^{\\alpha} (\\rho^{|\\alpha|}_{\\circ} f)\\|_{K^{n-|\\alpha|}_{p,\\theta}(\\cD)} \\leq N \\|f\\|_{K^{n}_{p,\\theta}(\\cD)},\n$$\nand\n$$\n\\|f_{x}\\|_{K^{n-1}_{p,\\theta}(\\cD)}\\leq N\\|f\\|_{K^{n}_{p,\\theta-p}(\\cD)},\n$$\nwith $N$ independent of $f$.\n\\end{lemma}\n\n To formulate our conditions on the different parts of the equations, we will use the $L_p$-spaces of predictable stochastic processes on $\\Omega_T:=\\Omega\\times(0,T]$ taking values in the weighted Sobolev spaces introduced above. \nFor $p>1$, $\\theta\\in{\\mathbb R}}\\def\\RR {{\\mathbb R}$, and $n\\in\\{0,1,2,\\ldots\\}$, we abbreviate \n$$\n\\ensuremath{\\mathbb{\\wso}}^{n}_{p,\\theta}(\\cD,T)\n:=\nL_p(\\Omega_T, \\ensuremath{\\mathcal{P}},\\ensuremath{\\mathbb P}\\otimes dx;\\ensuremath{K}^{n}_{p,\\theta}(\\cD)),\n$$\n$$\n \\ensuremath{\\mathbb{\\wso}}^{n}_{p,\\theta}(\\cD,T;\\ell_2)\n:=\nL_p(\\Omega_T, \\ensuremath{\\mathcal{P}},\\ensuremath{\\mathbb P}\\otimes dx;\\ensuremath{K}^{n}_{p,\\theta}(\\cD;\\ell_2)),\n$$\n$$\n\\bL^{[\\circ]}_{p,\\theta}(\\cD,T):=\\ensuremath{\\mathbb{\\wso}}^0_{p,\\theta}(\\cD,T), \\quad \\bL^{[\\circ]}_{p,\\theta}(\\cD,T;\\ell_2)\n:=\\ensuremath{\\mathbb{\\wso}}^0_{p,\\theta}(\\cD,T;\\ell_2),\n$$\nand\n\\[\n\\mathring{\\ensuremath{\\mathbb{\\wso}}}^1_{p,\\theta}(\\cD,T)\n:=\nL_p(\\Omega_T, \\ensuremath{\\mathcal{P}},\\ensuremath{\\mathbb P}\\otimes dx;\\mathring{\\ensuremath{K}}^{1}_{p,\\theta}(\\cD)).\n\\]\n\nUsing these spaces we introduce the following classes of stochastic processes that are tailor-made for the analysis of Equation~\\eqref{eq:SHE:Intro} on $\\cD$.\n\n\\begin{defn}\n \\label{defn sol}\n For $p\\geq 2$ and $\\theta\\in \\bR$\nwe write $u\\in\\mathcal{K}^1_{p,\\theta,0}(\\cD,T)$ if\n$u\\in\\mathring{\\ensuremath{\\mathbb{\\wso}}}^1_{p,\\theta-p}(\\cD,T)$\n and\nthere exist $f^0\\in \\bL^{[\\circ]}_{p,\\theta+p}(\\cD,T), f^i\\in \\bL^{[\\circ]}_{p,\\theta}(\\cD,T)$, $i=1,2$, and $g\\in \\bL^{[\\circ]}_{p,\\theta}(\\cD,T;\\ell_2)$,\nsuch that \n\\begin{equation}\\label{eqn 28}\n du=(f^0+f^i_{x^i})\\, dt +g^k \\, dw^k_t,\\quad t\\in(0,T],\n\\end{equation}\n on $\\cD$ in the sense of distributions with $u(0,\\cdot)=0$, that is, for any $\\varphi \\in\n\\ensuremath{\\mathcal{C}}^{\\infty}_{c}(\\cD)$, with probability one, the equality\n\\begin{equation}\\label{eq:distribution}\n(u(t,\\cdot),\\varphi)= \\int^{t}_{0}\n\\left[(f^0(s,\\cdot),\\varphi) -(f^i(s,\\cdot),\\varphi_{x^i}) \\right]ds + \\sum^{\\infty}_{k=1} \\int^{t}_{0}\n(g^k(s,\\cdot),\\varphi)\\, dw^k_s\n\\end{equation}\nholds for all $t \\leq T$. \nIn this situation\nwe also write\n$$\n\\bD u:=f^0+f^i_{x^i}\\qquad\\text{and}\\qquad \\bS u :=g\n$$\nfor the deterministic part and the stochastic part, respectively.\n\\end{defn}\n\n\n\n\\begin{remark}\nThe spaces $\\cK^1_{p,\\theta,0}(\\cD,T)$ from Definition~\\ref{defn sol} coincide with the spaces $\\mathfrak{K}^1_{p,\\theta}(\\cD,T)$ introduced in~\\cite[Definition~3.4]{CioKimLee+2018}.\nThe only (apparent) difference is that in the definition of $\\mathfrak{K}^1_{p,\\theta}(\\cD,T)$ the deterministic part $\\bD u$ is required to be an element of $\\mathbb K^{-1}_{p,\\theta+p}(\\cD,T):=L_p(\\Omega_T;K^{-1}_{p,\\theta+p}(\\cD))$, where $K^{-1}_{p,\\theta+p}(\\cD)$ is the dual of $\\mathring{K}^1_{p',\\theta'-p'}(\\cD)$ with $1\/p+1\/p'=1$ and $\\theta\/p+\\theta'\/p'=2$.\nHowever, this is not really a difference, since \n$$\n\\mathbb K^{-1}_{p,\\theta+p}(\\cD,T)=\\ggklam{f^0+f^i_{x^i} : f^0\\in \\bL^{[\\circ]}_{p,\\theta+p}(\\cD,T), f^i \\in \\bL^{[\\circ]}_{p,\\theta}(\\cD,T)}.\n$$\nThis can be proven with a similar strategy as the analogous result for classical Sobolev spaces, see, e.g., \\cite[page~62ff.]{AdaFou2003}, and by using the fact that for reflexive Banach spaces $B$ with dual $B^*$, the dual of $L_{p'}(\\Omega_T;B)$ is isometrically isomorphic to $L_p(\\Omega_T;B^*)$, see, e.g., \\cite[Theorem~IV.1.1 and Corollary~III.2.13]{DieUhl1977}.\n\n\\end{remark}\n\n In this article, Equation~\\eqref{eq:SHE:Intro} has the following meaning on $\\cD$.\n\n\n\n\\begin{defn}\\label{def:solution:D}\nWe say that $u$ is a solution to Equation~\\eqref{eq:SHE:Intro} on $\\cD$ in the class $\\mathcal{K}^{1}_{p,\\theta,0}(\\cD,T)$ \nif\n$u\\in \\mathcal{K}^{1}_{p,\\theta,0}(\\cD,T)$ with\n\\[\n\\bD u = \\Delta u + f^0+f^i_{x^i}=f^0+(f^i+u_{x^i})_{x^i}\n\\qquad\n\\text{and}\n\\qquad\n\\bS u = g.\n\\]\n\\end{defn}\n\n\nNow that we have specified the setting, we are ready to present our results. We start with the key estimate in this article. Its proof is given in Section~\\ref{sec:proof:lift}.\nRecall that $\\rho(x):=\\rho_{\\cD}(x):=\\mathrm{dist}(x,\\partial\\cD)$ denotes the distance of a point $x\\in\\cD$ to the boundary $\\partial \\cD$.\n\n\n\n \n\\begin{thm}\\label{lem:estim:2DCone:Theta}\nLet $p\\ge 2$, $1<\\Theta0$ depend only on $\\kappa_0$ and $\\lambda$. \nSince $\\theta$ satisfies~\\eqref{eq:range:vertex}, we can take $\\lambda$ sufficiently large such that $1-\\lambda<\\theta\/p<1+\\lambda$. Then the kernel\n$$\n\\cT_1(t,s,x,y):=\\ensuremath{\\mathds 1}_{x\\in \\cD} \\ensuremath{\\mathds 1}_{y\\in \\cD} \\ensuremath{\\mathds 1}_{t>s} |x|^{-1} \\frac{|x|^{(\\theta-2)\/p}}{|y|^{(\\theta-2)\/p}}\\Gamma_y(t-s,x,y)\n$$\nsatisfies the algebraic conditions in \\cite[Proposition~A.5]{KozNaz2014} with $\\mu=(\\theta-2)\/p$, $\\lambda_1=\\lambda_2=\\lambda-1$ and $r=1$. Hence by this proposition, \n\\[\n\\|v\\|_{\\bL^{[\\circ]}_{p,\\theta-p}(\\cD,T)}=\\|\\rho_{\\circ}^{-1}v\\|_{\\bL^{[\\circ]}_{p,\\theta}(\\cD,T)}\\leq N(p,\\theta,\\kappa_0)\\sum_i \\|f^i\\|_{\\bL^{[\\circ]}_{p,\\theta}(\\cD,T)}.\n\\]\n\n\\noindent\\emph{Step 2.} Assume the $f^i$s are sufficiently nice, say, $f^i\\in L_p(\\Omega_T, \\ensuremath{\\mathcal{P}}; \\ensuremath{\\mathcal{C}}^2_c(\\cD))$. Then by \\cite[Theorem~3.7]{CioKimLee+2018},\n$$\nv:=\\sum_i \\int_0^t \\int_\\cD \\Gamma (t-s,x,y) f^i_{x^i}(s,y)\\,dy\\,ds=-\\sum_i \\int_0^t \\int_\\cD \\Gamma_{y^i}(t-s,x,y) f^i(s,y)\\,dy\\,ds\n$$\nis the unique solution to Equation~\\eqref{eqn 4.2.1} in the class $\\cK^{1}_{p,\\theta,0}(\\cD,T)$, see also~\\cite{Naz2001, Sol2001}.\nThis, together with Step~1 and Theorem~\\ref{lem:estim:2DCone:Theta} with $m=0$ and $\\Theta=2$ lead to \\eqref{eqn 4.2.3} for $f^i\\in L_p(\\Omega_T, \\ensuremath{\\mathcal{P}}; \\ensuremath{\\mathcal{C}}^2_c(\\cD))$. \n\n\\noindent\\emph{Step 3.} General $f^i\\in \\bL^{[\\circ]}_{p,\\theta}(\\cD,T)$, $i=1,2$. Uniqueness follows from the case $f^i=0$. Take a sequence $(f^i_n)_{n\\in\\bN}\\subset L_p(\\Omega_T, \\ensuremath{\\mathcal{P}}; \\ensuremath{\\mathcal{C}}^2_c(\\cD))$ such that\n $f^i_n \\to f^i$ in $\\bL^{[\\circ]}_{p,\\theta}(\\cD,T)$ for each $i$. Let $v_n \\in \\cK^{1}_{p,\\theta,0}(\\cD,T)$ be the solution to Equation~\\eqref{eqn 4.2.1}\n with $f^i_n$. Then by Step~1 and Step~2, $(v_n)$ is a Cauchy sequence in $\\mathring{\\ensuremath{\\mathbb{\\wso}}}^1_{p,\\theta-p}(\\cD,T)$. \nLet $u:=\\lim_{n\\to \\infty} v_n$ in $\\mathring{\\ensuremath{\\mathbb{\\wso}}}^1_{p,\\theta-p}(\\cD,T)$. Fix $\\varphi \\in \\ensuremath{\\mathcal{C}}^{\\infty}_c(\\cD)$. \nThen taking the limit in\n $$\n (v_n(t,\\cdot), \\varphi)=-\\sum_i \\int^t_0((v_n(s,\\cdot))_{x^i}+f^i_n(s,\\cdot), \\varphi_{x^i})ds, \\quad \\forall\\; t\\leq T, \\quad (\\ensuremath{\\mathbb P}\\textup{-a.s.})\n $$\nand using the continuity of $t\\mapsto (u(t),\\varphi)$ (due to Estimate~\\eqref{eq:estim:sup:2DCone} from Lemma~\\ref{lem 4.5.1}), we find that $du=(\\Delta u+f^i_{x^i})\\, dt$ in the sense of distributions. The integral representation formula for $u$ is due to the fact that by Step~1 we also know that $\\lim_{n\\to\\infty}v_n=v$ in $L_{p,\\theta-p}^{[\\circ]}(\\cD,T)$. Estimate~\\eqref{eqn 4.2.3} follows by taking the limits in the estimates for $v_n$ proven in Step~2. \n\\end{proof}\n\n\\begin{remark}\nSince Lemma~\\ref{lem:fi:estim:2DCone} addresses the deterministic heat equation, the restriction $p\\geq 2$ is obsolete. The result as well as the proof carry over to the case $p>1$ mutatis mutandis.\n\\end{remark}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:ex:uni:2DCone}]\nThis is now an immediate consequence of \\cite[Theorem~3.7]{CioKimLee+2018} and Lemma~\\ref{lem:fi:estim:2DCone} above.\n\\end{proof}\n\nTheorem \\ref{lem:estim:2DCone:Theta} with $\\Theta=2$ and Estimate~\\eqref{lem:estim:2DCone:Theta} now lead to the following higher order regularity result of the solution depending on the regularity of the forcing terms $f^{0}$, $f^i$, and $g^k$. \n Recall that in this section $\\rho$ denotes the distance to the boundary of $\\cD$.\n\n\\begin{corollary}[higher order regularity\/angular domains]\n\\label{high}\n Given the setting of Theorem~\\ref{thm:ex:uni:2DCone}, let $u$ be the unique solution in the class $\\cK^{1}_{p,\\theta,0}(\\cD,T)$ to Equation~\\eqref{eq:SHE:Intro} on $\\cD$. Assume that \n\\begin{align*}\nC(m,\\theta, f^i,f^0,g)\n&:=\n{\\mathbb E} \\int^T_0 \\int_{\\cD}\\ssgrklam{\\sum_{|\\alpha|\\leq (m-1)\\vee 0} |\\rho^{\\abs{\\alpha}+1}D^{\\alpha}f^0|^p+\n \\sum_i\\sum_{|\\alpha|\\leq m} |\\rho^{|\\alpha|}D^{\\alpha}f^i|^p\\\\\n&\\quad\\quad\\qquad\\qquad\\qquad+|\\,\\rho_{\\circ} f^0|^p+\n\\sum_{\\abs{\\alpha}\\leq m} |\\rho^{\\abs{\\alpha}}D^\\alpha g|_{\\ell_2}^p }\\rho_{\\circ}^{\\theta-2}\\, dx\\,dt <\\infty\n\\end{align*}\nfor some $m\\in\\{0,1,2,\\ldots\\}$. Then\n$$\n{\\mathbb E}\\int_0^T \\sum_{\\abs{\\alpha}\\leq m+1} \\int_{\\cD} \\Abs{\\rho^{\\abs{\\alpha}-1}D^\\alpha u}^p \\abs{x}^{\\theta-2}\\,dx\\,dt\\leq N\\,C(m,\\theta, f^i,f^0,g)<\\infty,$$\nwhere $N=N(p,\\theta,\\kappa_0,m)$. In particular, $N$ does not depend on $T$.\n\n\\end{corollary}\n\n\nWe will need the following `general uniqueness' lemma to handle the stochastic heat equation on polygons in Section~\\ref{sec:Polygons}.\n\n\\begin{lemma}\n \\label{lem for uniqueness} \nLet $2\\leq p_1\\leq p_2$ and let $\\theta_1, \\theta_2\\in\\bR$ satisfy \\eqref{eq:range:vertex} for $p=p_1$ and $p=p_2$, respectively. Assume for both $j=1$ and $j=2$,\n\\[\nf^0 \\in \\bL^{[\\circ]}_{p_j,\\theta_j+p_j}(\\cD,T), \\quad f^i \\in \\bL^{[\\circ]}_{p_j,\\theta_j}(\\cD,T),\\,\\,i=1,2, \\quad g\\in \\bL^{[\\circ]}_{p_j,\\theta_j}(\\cD,T;\\ell_2)\n\\]\n and let $u\\in \\cK^{1}_{p_1,\\theta_1,0}(\\cD,T)$ be the solution to Equation~\\eqref{eq:SHE:Intro}. Then $u\\in \\cK^{1}_{p_2,\\theta_2,0}(\\cD,T)$. \\end{lemma}\n \n\\begin{proof}\nThis follows from the integral representation formula of the solution in Theorem \\ref{thm:ex:uni:2DCone}, that is, the unique solutions in \n$\\cK^{1}_{p_1,\\theta_1,0}(\\cD,T)$ and $\\cK^{1}_{p_2,\\theta_2,0}(\\cD,T)$ have the same representation formula.\n\\end{proof}\n\n\n\n\n\n\\begin{remark}\n\\label{remark cones}\n To keep the presentation short, the results in this section are formulated only for angular domains $\\cD\\subseteq\\bR^2$ with vertex at the origin and with one of the edges being the positive $x^1$-axis.\nHowever, since every angular domain in $\\bR^2$ can be seen as a translation of a rotation of such a domain, all results can be extended accordingly, as the Laplace operator is invariant under translations and rotations. \nMore precisely, fix $a\\in (-\\pi, \\pi)$ and $x_0\\in \\bR^2$. Let\n \\begin{equation*}\n\\tilde{\\cD}:=\\tilde{\\cD}_{\\kappa_0}(x_0,a):=\\big\\{x\\in \\bR^2: x=x_0+(r\\cos\\vartheta,r\\sin\\vartheta),\\; r>0,\\;\\vartheta\\in (a,a+\\kappa_0)\\big\\}.\n\\end{equation*}\nReplacing $\\cD$ and $\\rho_\\circ$ by $\\tilde{\\cD}_{\\kappa_0}(x_0,a)$ and $\\tilde{\\rho}_\\circ(x):=|x-x_0|$, respectively, in the definitions of the weighted Sobolev spaces from above, we can define analogous spaces, such as $K^n_{p,\\theta}(\\tilde{\\cD})$, $\\mathbb K^n_{p,\\theta}(\\tilde{\\cD},T)$ and \n$\\cK^1_{p,\\theta,0}(\\tilde{\\cD},T)$, on $\\tilde{\\cD}$. \nThen, the results in this section hold with $\\tilde{\\cD}$ in place of $\\cD$. Indeed,\nlet $Q=(q_{ij})_{1\\leq i,j\\leq 2}$ be the orthogonal matrix such that $\\tilde{\\cD}_{\\kappa_0}(x_0,a)=x_0+Q \\cD_{\\kappa_0}$. Then, since the Laplacian is invariant under the rotations and translations, the statement that $u\\in \\cK^1_{p,\\theta,0}(\\tilde{\\cD},T)$ satisfies\n\\begin{equation}\n\\label{eqn 4.10.7}\ndu=(\\Delta u +f^0+f^i_{x^i})\\,dt+g^k dw^k_t,\n\\end{equation}\n in the sense of distribution (analogous meaning to Definition~\\ref{defn sol}) is the same as the statement that $v(t,x):=u(t,x_0+Qx)\\in \\cK^1_{p,\\theta,0}(\\cD,T)$ satisfies\n\\[\ndv=(\\Delta v +\\tilde{f}^0+\\tilde{f}^i_{x^i})\\, dt+\\tilde{g}^k \\, dw^k_t,\n\\]\nwhere\n$\n\\tilde{f}^0(t,x)=f^0(t,x_0+Qx)$, $\\tilde{f}^i(t,x)=q_{1i}f^1(t,x_0+Qx)+q_{2i}f^2(t,x_0+Qx)$, $i=1,2$,\nand $\\tilde{g}(t,x)=g(t,x_0+Qx)$. \nHence, all existence and uniqueness results as well as all estimates can be extended to general angular domains, since, obviously,\n$$\n\\|h(x)\\|_{K^n_{p,\\theta}(\\tilde{\\cD})} \\sim \\|h(x_0+Qx)\\|_{K^n_{p,\\theta}(\\cD)}\n$$\nfor any $h\\in K^n_{p,\\theta}(\\tilde{\\cD})$. \n To extend Lemma~\\ref{lem 4.5.1}, formally set $\\Delta u=0$ in~\\eqref{eqn 4.10.7}.\n\\end{remark}\n\n\\mysection{Proof of Theorem \\ref{lem:estim:2DCone:Theta}}\\label{sec:proof:lift}\n\n\nIn this section we give a detailed proof of the key estimate from Theorem~\\ref{lem:estim:2DCone:Theta}.\nOur proof is based on a suitable a-priori estimate for the stochastic heat equation on $\\ensuremath{\\mathcal{C}}^1$ domains, as presented in Lemma~\\ref{lem 10} below.\nWe use this result to establish an estimate for the solution on a subdomain of $\\cD$ which is bounded away from the vertex and from infinity (see Lemma~\\ref{lem:estim:2DCone:U1} below).\nThen we can prove Theorem~\\ref{lem:estim:2DCone:Theta} by using a dilation argument, as $\\cD$ is invariant under positive dilation.\nFor this strategy to succeed, it is crucial that the constant in Lemma~\\ref{lem 10} does not depend on the time horizon $T$.\n\n\n\n\nWe start with the definition of the weighted Sobolev spaces $H^n_{p,\\Theta}(G)$ on $\\ensuremath{\\mathcal{C}}^1$ domains $G\\subseteq\\bR^d$ ($d\\geq 1$), which we need for the statement of Lemma~\\ref{lem 10}. First we recall the definition of a $\\ensuremath{\\mathcal{C}}^1$ domain. \n\n\\begin{defn}\\label{definition domain}\n Let $G$ be a domain in $\\bR^d$, $d\\geq 1$.\nWe write $\\partial G\\in \\ensuremath{\\mathcal{C}}^1_u$ and say that $G$ is a $\\ensuremath{\\mathcal{C}}^1$ domain if there exist constants $r_0, K_0\\in(0,\\infty)$ such that \nfor any $x_0 \\in \\partial G$ there exists\n a one-to-one continuously differentiable mapping $\\Psi$ of\n $B_{r_0}(x_0)$ onto a domain $J\\subset\\bR^d$ such that\n\\begin{enumerate}[align=right,label=\\textup{(\\roman*)}] \n\\item $J_+:=\\Psi(B_{r_0}(x_0) \\cap G) \\subset \\bR^d_+$ and\n$\\Psi(x_0)=0$;\n\n\\item $\\Psi(B_{r_0}(x_0) \\cap \\partial G)= J \\cap \\{y\\in\n\\bR^d:y^1=0 \\}$;\n\n\\item $\\|\\Psi\\|_{\\ensuremath{\\mathcal{C}}^{1}(B_{r_0}(x_0))} \\leq K_0 $ and\n$|\\Psi^{-1}(y_1)-\\Psi^{-1}(y_2)| \\leq K_0 |y_1 -y_2|$ for any $y_i\n\\in J$;\n\n\\item $\\Psi_x$ is uniformly continuous in $B_{r_0}(x_0)$.\n\n \\end{enumerate}\n\\end{defn}\nThroughout this article, we assume that $G$ is either $\\bR^d_+:=\\{x\\in \\bR^d\\colon x^1>0\\}$ or a bounded $\\ensuremath{\\mathcal{C}}^1$ domain in $\\bR^d$ ($d\\geq 1$). Note that in both cases, $G$ is of class $\\ensuremath{\\mathcal{C}}^1_u$ in the sense of \\cite[Assumption~2.1]{Kim2004}.\nRecall that $\\rho(x)=\\rho_G(x)=\\mathrm{dist}(x,\\partial G)$ for $x\\in G$;\n$\\rho(x)=x^1$ if $G=\\bR^2_+$. \nFor $p>1$ and $\\Theta\\in\\bR$, we write\n\\[\nL_{p,\\Theta}(G):=L_{p}(G,\\rho^{\\Theta-d} dx;{\\mathbb R}}\\def\\RR {{\\mathbb R}) \\quad \n\\text{and}\n\\quad\nL_{p,\\Theta}(G;\\ell_2):=L_p(G,\\rho^{\\Theta-d} dx;\\ell_2)\n\\]\nfor the weighted $L_p$-spaces of real-valued\/$\\ell_2$-valued functions with weight $\\rho^{\\Theta-d}$.\nFor $n\\in \\{0,1,2,\\ldots\\}$, by $H^n_{p,\\theta}(G)$ we denote the space of all \n$f\\in L_{p,\\Theta}(G)$ such that\n\\begin{equation}\n \\label{eqn 4.9.5}\n\\|f\\|^p_{H^n_{p,\\Theta}(G)}:=\\sum_{|\\alpha|\\leq n} \\|\\rho^{\\abs{\\alpha}} D^\\alpha f\\|^p_{L_{p,\\Theta}(G)}<\\infty.\n\\end{equation}\nMoreover, we define the dual spaces\n\\[\nH^{-n}_{p,\\Theta}(G):=\\grklam{H^n_{p',\\Theta'}(G)}^*,\\qquad\\frac{1}{p}+\\frac{1}{p'}=1,\\quad \\frac{\\Theta}{p}+\\frac{\\Theta'}{p'}=d.\n\\]\nThe space $H^n_{p,\\Theta}(G;\\ell_2)$ is defined analogously for $n\\in \\bZ$.\n\nTo state the main properties of these spaces, we introduce some additional notation. \nFor $k\\in\\{0,1,2,\\ldots\\}$, let\n$$\n|f|^{(0)}_{k}:=|f|^{(0)}_{k,G} :=\\sup_{\\substack{x\\in G\\\\\n|\\beta| \\leq k}}\\rho^{|\\beta|}(x)|D^{\\beta}f(x)|.\n$$\nIf $G$ is bounded, let $\\psi$ be a bounded $\\ensuremath{\\mathcal{C}}^\\infty$ function defined in $G$ with $|\\psi|^{(0)}_k+|\\psi_x|^{(0)}_k<\\infty$ for any $k$, which is comparable to $\\rho$, i.e., $N^{-1}\\rho(x)\\leq \\psi(x)\\leq N \\rho(x)$ for some constant $N>0$; see, e.g., \\cite[ Section~2]{KimKry2004}.\nIt is known that, if $G$ is bounded, then the map $\\Psi$ in Definition~\\ref{definition domain} can be chosen in such a way that $\\Psi$ is infinitely differentiable in $B_{r_0}(x_0)\\cap G$ and for any multi-index $\\alpha$\n\\begin{equation}\n \\label{eqn 12.4.9}\n\\sup_i \\sup_{B_{r_0}(x_0) \\cap G} \\rho^{|\\alpha|}|D^{\\alpha}\\Psi_{x^i}|\\leq N(\\alpha)<\\infty;\n \\end{equation}\nsee, e.g., \\cite{KimKry2004} or the proof of \\cite[Lemma~4.9]{KimLee2011}.\nActually, after appropriate rotation and translation, one can take $\\Psi(x^1,x')=(\\psi(x),x')$. By \\cite[Theorem~3.2]{Lot2000} and \\eqref{eqn 12.4.9} above, if $\\text{supp}\\, u\\subset B_{r}(x_0)\\cap \\overline{G}$ and $r0$,\n$$\nN^{-1}\\|u\\|_{H^n_p(\\bR^d)}\\leq \\|u\\|_{H^n_{p,\\Theta}(G)}\\leq N\\|u\\|_{H^n_p(\\bR^d)},\n$$\nwhere $H^n_p(\\bR^d):=\\{u: D^{\\alpha}u\\in L_p(\\bR^d), \\, \\forall\\, |\\alpha|\\leq n\\}$ if $n\\geq 0$, and otherwise it is the dual space of $H^{-n}_q(\\bR^d)$, where $\\frac1p+\\frac1q=1$.\n\\end{enumerate}\n\n\\end{lemma}\n\nNote that, by Lemma~\\ref{collection}\\ref{col:multiplier} and the properties of $\\psi$, $\\psi$ is a point-wise multiplier in $H^n_{p,\\Theta}(G)$ if $G$ is bounded.\n\n For the corresponding spaces of predictable $H^n_{p,\\Theta}(G)$\/$H^n_{p,\\Theta}(G;\\ell_2)$-valued stochastic processes we use the abbreviations\n$$\n\\bH^{n}_{p,\\Theta}(G,T)\n:=\nL_p(\\Omega_T, \\ensuremath{\\mathcal{P}};H^{n}_{p,\\Theta}(G))\n\\quad \n\\text{and}\\quad\\bL_{p,\\Theta}(G,T):=\\bH^0_{p,\\Theta}(G,T),\n$$\nas well as\n\\[\n\\bH^n_{p,\\Theta}(G,T;\\ell_2)\n:=\nL_p(\\Omega_T, \\ensuremath{\\mathcal{P}};H^{n}_{p,\\Theta}(G;\\ell_2))\\quad \n\\text{and}\\quad\\bL_{p,\\Theta}(G,T;\\ell_2):=\\bH^0_{p,\\Theta}(G,T;\\ell_2).\n\\]\n The following classes of stochastic processes are tailor-made for the analysis of Equation~\\eqref{eq:SHE:Intro} on $G$.\n\\begin{defn}\nFor $p\\geq 2$ and $\\Theta\\in\\bR$ we write $u\\in\\frH^n_{p,\\Theta,0}(G,T)$ if\n$u\\in\\bH^n_{p,\\Theta-p}(G,T)$\n and\nthere exist \n$f\\in \\bH^{n-2}_{p,\\Theta+p}(G,T)$ and\n $g\\in \\bH^{n-1}_{p,\\Theta}(G,T;\\ell_2)$\nsuch that \n\\begin{equation*}\\label{eqn 28_1}\n du=f\\, dt +g^k \\, dw^k_t,\\quad t\\in (0,T],\n\\end{equation*}\non $G$ in the sense of distributions with $u(0,\\cdot)=0$; see Definition~\\ref{defn sol} accordingly. We denote \n$$\n\\bD u:=f\\qquad\\text{and}\\qquad \\bS u :=g.\n$$\n\n\\end{defn}\n\n In this article, Equation~\\eqref{eq:SHE:Intro} has the following meaning on $G$.\n\\begin{defn}\\label{defn sol:C1}\nWe say that $u\\in\\bH^{n}_{p,\\Theta-p}(G,T)$ is\na solution to Equation~\\eqref{eq:SHE:Intro} on $G$ \nin the class $\\frH^{n}_{p,\\Theta,0}(G,T)$\nif\n$u\\in \\frH^{n}_{p,\\Theta,0}(G,T)$ with\n\\[\n\\bD u = \\Delta u + f^0+f^i_{x^i}\n\\qquad\n\\text{and}\n\\qquad\n\\bS u = g.\n\\]\n\\end{defn}\n\n\\begin{remark}\nAll definitions above are given only for $\\ensuremath{\\mathcal{C}}^1$ domains, as we say from the beginning that in this article $G$ is either a bounded $\\ensuremath{\\mathcal{C}}^1$ domain or the half plane. However, all the spaces defined above as well as the solution concept make sense on any domain $\\domain\\subset\\bR^d$ with non-empty boundary.\n\\end{remark}\n\n\n Now we have all notions we need in order to state and prove the a-priori estimate for Equation~\\eqref{eq:SHE:Intro} on bounded $\\ensuremath{\\mathcal{C}}^1$ domains that we use to prove Lemma~\\ref{lem:estim:2DCone:U1} and therefore Theorem~\\ref{lem:estim:2DCone:Theta}.\n\n\n\\begin{lemma}\\label{lem 10}\nLet $G\\subset\\bR^d$ be a bounded $\\ensuremath{\\mathcal{C}}^1$ domain, $p\\geq2$, $n\\in\\{-1,0,1,\\ldots\\}$, and $d-1< \\Theta0$ such that \n\\begin{equation}\\label{eqn 5.6.544}\n\\sum_{n=-\\infty}^{\\infty}\\eta(e^{n+t})>c>0, \\quad \\forall \\, t\\in \\bR.\n\\end{equation}\n\n\n\n\nOur proof of Lemma~\\ref{lem 4.5.1} relies on the the following characterization of the $L_{p,\\theta}^{[\\circ]}(\\cD)$-norm.\n\n\\begin{lemma} \n\\label{lem 3.1}\nLet $p>1$ and $\\theta\\in \\bR$. Let $u\\colon\\cD\\to\\bR$ be a measurable function. \n\\begin{enumerate}[leftmargin=*,label=\\textup{(\\roman*)}, wide] \n\\item\\label{lem 3.1.1} If $\\eta$ and $G$ are as above, then\n$$\n\\|u\\|^p_{L^{[\\circ]}_{p,\\theta}(\\cD)} \\sim \\sum_{n \\in \\bZ} e^{n\\theta} \\|\\eta(|x|) u(e^nx)\\|^p_{L_p(\\cD)}= \\sum_{n \\in \\bZ} e^{n\\theta} \\|\\eta(|x|) u(e^nx)\\|^p_{L_p(G)}.\n$$\n\\item\\label{lem 3.1.2} For any function $\\xi \\in C^{\\infty}_0((0,\\infty))$ we have\n$$\n\\sum_{n \\in \\bZ} e^{n\\theta} \\|\\xi(|x|) u(e^nx)\\|^p_{L_p(\\cD)}\\leq N(\\xi, \\eta, p,\\theta) \\|u\\|^p_{L^{[\\circ]}_{p,\\theta}(\\cD)}.\n$$\n\\end{enumerate}\n\\end{lemma}\n\n\n\n\\begin{proof}\nTo see~\\ref{lem 3.1.1}, it is enough to repeat the proof of \\cite[Remark~1.3]{Kry1999c}. Indeed, by the change of variables $e^nx \\to x$,\n$$\n\\sum_{n\\in \\bZ} e^{n\\theta}\\|\\eta(|x|)u(e^nx)\\|^p_{L_p(\\cD)}=\\int_{\\cD} \\zeta(x)|u(x)|^pdx,\n$$\nwhere\n$$\n\\zeta(x)=\\sum_{n\\in \\bZ} e^{n(\\theta-2)}\\eta^p(e^{-n}|x|) \\sim |x|^{\\theta-2},\n$$\n see \\cite[Remark~1.3]{Kry1999c}. Moreover, since $\\mathrm{supp}\\,\\eta\\cap\\cD\\subset G$, the equality in~\\ref{lem 3.1.1} is also satisfied. Part~\\ref{lem 3.1.2}\nholds since\n $$\n\\sum_{n\\in \\bZ} e^{n(\\theta-2)}\\xi^p(e^{-n}|x|) \\leq N(\\xi,\\eta,\\theta,p) |x|^{\\theta-2};\n$$\nsee \\cite[Lemma~1.4]{Kry1999c} for details.\n\\end{proof}\n\nIn addition to Lemma~\\ref{lem 3.1}, we also need the following counterpart of Lemma~\\ref{lem 4.5.1} for the stochastic heat equation on bounded $\\ensuremath{\\mathcal{C}}^1$ domains.\nIn the proof, we are going to use the common abbreviations\n\\[\n\\bH^n_p(T):=L_p(\\Omega_T, \\cP;H^n_p(\\bR^d)) \\quad\\text{and}\\quad\n\\bL_p(T;\\ell_2):=L_p(\\Omega_T,\\cP;L_p(\\bR^d;\\ell_2)),\n\\]\nfor $n\\in\\bZ$.\n\n\n\\begin{lemma}\n\\label{lem krylov}\nLet $G$ be a bounded $\\ensuremath{\\mathcal{C}}^1$ domain, $\\Theta\\in \\bR$, $p\\geq 2$, and $u\\in \\frH^{1}_{p,\\Theta,0}(G,T)$ with $du=fdt +g\\,dw^k_t$. Then $u\\in L_p(\\Omega; \\ensuremath{\\mathcal{C}}([0,T]; L_{p,\\Theta}(G))$, and for any $c>0$,\n\n\\begin{eqnarray*}\n{\\mathbb E} \\sup_{t\\leq T} \\|u(t,\\cdot)\\|^p_{L_{p,\\Theta}(G)}\\le N \\Big( c \\|u\\|^p_{\\bH^{1}_{p,\\Theta-p}(G,T)} \n + c^{-1}\\|f\\|^p_{\\bH^{-1}_{p,\\Theta+p}(G,T)}+\\|g\\|^p_{\\bL_{p,\\Theta}(G,T;\\ell_2)} \\Big),\n \\end{eqnarray*}\nwhere $N= N(d,p,\\theta,G, T)$. In particular, if $f=f^0+f^i_{x^i}$, then the right hand side above is bounded by a constant multiple of \n\\[\n c \\|u\\|^p_{\\bH^1_{p,\\Theta-p}(G,T)}+c^{-1}\\|f^0\\|^p_{\\bL_{p,\\Theta+p}(G,T)}+c^{-1}\\|f^i\\|^p_{\\bL_{p,\\Theta}(G,T)}\n +\\|g\\|^p_{\\bL_{p,\\Theta}(G,T;\\ell_2)}.\n\\]\n \\end{lemma}\n\n\n\n\\begin{proof}\nIntroduce a partition of unity $\\zeta_0, \\zeta_1,\\zeta_2,\\cdots, \\zeta_M$\n of $G$ such that $\\zeta_0\\in \\ensuremath{\\mathcal{C}}^{\\infty}_c(G)$ and\n $\\zeta_j\\in \\ensuremath{\\mathcal{C}}^{\\infty}_c(B_{r}(x_j))$ $(j=1,2,\\cdots,M)$, where\n $x_j\\in \\partial G$ and $r2$) and \\cite[Remark~4.14]{Kry2001} (for $p=2$), \n\\begin{equation}\\label{eq:cts:zeta0}\n\\zeta_0 u \\in L_p(\\Omega; \\ensuremath{\\mathcal{C}}([0,T]; L_p(\\bR^d)),\n\\end{equation}\nand there exists a constant $N$, such that, for any $c>0$,\n\\begin{align*}\n{\\mathbb E} \\sup_{t\\leq T} \\|\\zeta_0u\\|^p_{L_p}\n&\\leq N c \\|\\zeta_0u\\|^p_{\\bH^1_p(T)}+\nNc^{-1}\\|\\zeta_0 f\\|^p_{\\bH^{-1}_p(T)}+N \\|\\zeta_0g\\|^p_{\\bL_p(T;\\ell_2)}\\\\\n&\\leq Nc \\|\\zeta_0u\\|^p_{\\bH^1_{p,\\Theta-p}(G,T)}+Nc^{-1}\\|\\zeta_0f\\|^p_{\\bH^{-1}_{p,\\Theta}(G,T)}\n+N\\|\\zeta_0g\\|^p_{\\bL_{p,\\Theta}(G,T;\\ell_2)}.\n\\end{align*}\nMoreover, for every $j\\in\\{1,\\ldots,M\\}$,\n\\[\nd((\\zeta_j u)(\\Psi^{-1}_j))\n=\n(\\zeta_j f)(\\Psi^{-1}_j)\\,dt+(\\zeta_jg^k)(\\Psi^{-1}_j) \\,dw^k_t =:F_j \\,dt+G^k_j \\,dw^k_t, \\quad t\\in (0,T],\n\\]\non $\\bR^d_+$ and due to~\\eqref{relation}, $(\\zeta_ju)(\\Psi^{-1}_j) \\in \\frH^1_{p,\\Theta}(\\bR^d_+,T)$ (see Section~\\ref{sec:proof:lift} for notation). Therefore,\nby \\cite[Theorem~4.1]{Kry2001} (for $p>2$) and \\cite[Remark~4.5]{Kry2001} (for $p=2$),\n\\begin{equation}\\label{eq:cts:zetaj}\n(\\zeta_ju)(\\Psi^{-1}_j)\\in L_p(\\Omega; \\ensuremath{\\mathcal{C}}([0,T]; L_{p,\\Theta}(\\bR^d_+)),\n\\end{equation}\nand\n\\begin{align*}\n{\\mathbb E} &\\sup_{t\\leq T} \\|(\\zeta_ju)(\\Psi^{-1}_j)\\|^p_{L_{p,\\Theta}(\\bR^d_+)}\\\\\n&\\leq \nN c \\|(\\zeta_ju)(\\Psi^{-1}_j)\\|^p_{\\bH^1_{p,\\Theta-p}(\\bR^d_+,T)}\n+Nc^{-1}\\|F_j\\|^p_{\\bH^{-1}_{p,\\Theta+p}(\\bR^d_+,T)}+N\\|G_j\\|^p_{\\bL_{p,\\Theta}(\\bR^d_+,T;\\ell_2)}\\\\\n&\\leq N c \\|\\zeta_ju\\|^p_{\\bH^1_{p,\\Theta-p}(G,T)}\n+Nc^{-1}\\|\\zeta_jf\\|^p_{\\bH^{-1}_{p,\\Theta+p}(G,T)}+\nN\\|\\zeta_jg\\|^p_{\\bL_{p,\\Theta}(G,T;\\ell_2)}.\n\\end{align*}\nSumming up gives the desired estimate and $u\\in L_p(\\Omega;\\ensuremath{\\mathcal{C}}([0,T];L_{p,\\Theta}(G)))$ follows from~\\eqref{eq:cts:zeta0} and \\eqref{eq:cts:zetaj}, together with~\\eqref{eqv}.\nThe second assertion is due to the fact that, by Lemma~\\ref{collection}, \n\\[\n\\|f^i_{x^i}\\|_{H^{-1}_{p,\\Theta+p}(G)}\\leq N \\|\\psi f^i_{x^i}\\|_{H^{-1}_{p,\\Theta}(G)}\\leq N \\|f^i\\|_{L_{p,\\Theta}(G)}.\\qedhere\n\\]\n\\end{proof} \n\n\n\n\nWe have now all ingredients we need in order to prove Lemma~\\ref{lem 4.5.1}.\n\n\n\\begin{proof}[Proof of Lemma~\\ref{lem 4.5.1}]\nWe first prove Estimate~\\eqref{eq:estim:sup:2DCone}. By Lemma~\\ref{lem 3.1},\n\\begin{equation}\\label{eqn 2018-4}\n{\\mathbb E} \\sup_{t\\leq T} \\|u(t,\\cdot)\\|^p_{L^{[\\circ]}_{p,\\theta}(\\cD)} \n\\leq \nN \\sum_{n\\in \\bZ} e^{n\\theta} \\,{\\mathbb E} \\sup_{t\\leq T} \\|u(t,e^nx) \\eta(x)\\|^p_{L_p(G)}.\n\\end{equation}\n For $n\\in\\bZ$, let $v_n(t,x):=u(t,e^nx)\\eta(x)$. Then \n$$\ndv_n=[ e^{-n} (f^i(t,e^nx))_{x^i}\\eta(x)+f^0(t,e^nx)\\eta(x)]dt+g^k(t,e^nx)\\eta(x)dw^k_t, \\quad t\\in(0,T],\n$$\non $G$.\nNote that\n$$\n e^{-n} (f^i(t,e^nx))_{x^i}\\eta(x)= e^{-n} [f^i(t,e^nx)\\eta(x)]_{x^i}-e^{-n}f^i(t,e^nx)\\eta_{x^i}(x),\n$$\nand\n\\begin{equation}\\label{eq_sec4_2}\n(v_n)_{x^i}=e^{n}u_{x^i}(t,e^nx)\\eta(x)-u(t,e^nx)\\eta_{x^i}(x).\n\\end{equation}\nObviously, $L_{p,2}(G)=L_p(G)$ and by Hardy's inequality,\n\\begin{equation}\\label{eq_sec4_1}\n\\|v_n\\|_{H^1_{p,2-p}(G)}\n\\le \nN\\sgrklam{\\|\\rho^{-1}_G v_n\\|_{L_p(G)}+\\sum_i\\|(v_{n})_{x^i}\\|_{L_p(G)} }\n\\leq \nN \\|(v_{n})_{x}\\|_{L_p(G)}.\n\\end{equation}\nBy Lemma~\\ref{lem krylov} with $\\Theta=d=2$, \\eqref{eq_sec4_1}, and \\eqref{eq_sec4_2}, for any $c>0$\n\\begin{align*}\n&{\\mathbb E} \\sup_{t\\leq T} \\|v_n(t,\\cdot)\\|^p_{L_p(G)} \\\\\n&\\leq N \\Big(ce^{np}\\sum_i\\|u_{x^i}(\\cdot,e^n\\cdot)\\eta\\|^p_{\\bL_{p,d}(G,T)}+\nc\\sum_i\\|u(\\cdot,e^n\\cdot)\\eta_{x^i}\\|^p_{\\bL_{p,d}(G,T)} \\\\\n\\\\&\\quad \\quad +e^{-np}c^{-1}\\|f^i(\\cdot,e^n\\cdot)\\eta\\|^p_{\\bL_{p,d}(G,T)}\n+e^{-np}c^{-1}\\sum_i\\| \\rho f^i(\\cdot,e^n\\cdot)\\eta_{x^i}\\|^p_{\\bL_{p,d}(G,T)}\\\\\n&\\quad \\quad +c^{-1}\\|\\rho f^0(\\cdot,e^n\\cdot)\\eta\\|^p_{\\bL_{p,d}(G,T)} +\\|\\eta g(\\cdot,e^n\\cdot)\\|^p_{\\bL_{p,d}(G,T;\\ell_2)} \\Big).\n\\end{align*}\nSince $\\rho$ is bounded in $G$, we can drop $\\rho$ above, so that, if we choose\n$c:=e^{-np}$ and use~\\eqref{eqn 2018-4}, we get\n\\begin{align*}\n{\\mathbb E} &\\sup_{t\\leq T} \\|u(t,\\cdot)\\|^p_{L^{[\\circ]}_{p,\\theta}(\\cD)} \\\\\n&\\leq N \n\\ssgrklam{\\sum_n e^{n\\theta}\\|u_x(\\cdot,e^n\\cdot)\\eta\\|^p_{\\bL_{p,d}(\\cD,T)}\n+ \\sum_n e^{n(\\theta-p)}\\sum_i\\|u(\\cdot,e^n\\cdot)\\eta_{x^i}\\|^p_{\\bL_{p,d}(\\cD,T)}\\\\\n&\\quad\\quad\\quad+ \\sum_{n,i} e^{n\\theta}\\|f^i(\\cdot,e^n\\cdot)\\eta\\|^p_{\\bL_{p,d}(\\cD,T)}\n+ \\sum_{n,i} e^{n\\theta} \\|f^i(\\cdot,e^n\\cdot)\\eta_{x^i}\\|^p_{\\bL_p(\\cD,T)}\\\\\n&\\quad\\quad\\quad+ \\sum_n e^{n(\\theta+p)}\\|f^0(\\cdot,e^n\\cdot)\\eta\\|^p_{\\bL_{p,d}(\\cD,T)}\n+ N \\sum_n e^{n\\theta}\\|g(\\cdot,e^n\\cdot)\\eta\\|^p_{\\bL_{p,d}(\\cD,T;\\ell_2)}}.\n\\end{align*}\nTherefore, due to Lemma~\\ref{lem 3.1}\\ref{lem 3.1.2}, Estimate~\\eqref{eq:estim:sup:2DCone} holds.\n\nTo prove the continuity assertion, we take a sequence of smooth functions $\\xi_n\\in \\ensuremath{\\mathcal{C}}^{\\infty}_c(\\bR^2)$ such that $\\xi_n=1$ if $3\/n<|x|1$ and $n\\in \\{0,1,2,\\ldots\\}$, we define the spaces $K^n_{p,\\theta}(\\cO)$, $K^n_{p,\\theta}(\\cO;\\ell_2)$, $L^{[\\circ]}_{p,\\theta}(\\cO)$, and \n$L^{[\\circ]}_{p,\\theta}(\\cO;\\ell_2)$ in the same way as the corresponding spaces on $\\cD$ from Section~\\ref{sec:2DCone} with $\\rho_\\circ$ replaced by $\\tilde{\\rho}$, i.e., for instance,\n$$\n\\|u\\|^p_{K^n_{p,\\theta}(\\cO)}=\\sum_{|\\alpha|\\leq n} \\int_{\\cO} |\\tilde{\\rho}^{|\\alpha|} D^{\\alpha}u|^p \\tilde{\\rho}^{\\theta-2}dx.$$\nThe space $\\mathring{\\ensuremath{K}}^1_{p,\\theta}(\\cO)$ is the closure of the space $\\ensuremath{\\mathcal{C}}^\\infty_c(\\cO)$ of test functions in $\\ensuremath{K}^1_{p,\\theta}(\\cO)$. \nIn analogy to Section~\\ref{sec:2DCone}, for the $L_p$-spaces of predictable stochastic processes with values in the weighted Sobolev spaces introduced above we use the abbreviations \n\\[\n\\ensuremath{\\mathbb{\\wso}}^{n}_{p,\\theta}(\\cO,T)\n:=\nL_p(\\Omega_T, \\ensuremath{\\mathcal{P}};\\ensuremath{K}^{n}_{p,\\theta}(\\cO)),\n\\qquad \\ensuremath{\\mathbb{\\wso}}^{n}_{p,\\theta}(\\cO,T;\\ell_2)\n:=\nL_p(\\Omega_T, \\ensuremath{\\mathcal{P}};\\ensuremath{K}^{n}_{p,\\theta}(\\cO;\\ell_2)),\n\\]\n\\[\n\\bL^{[\\circ]}_{p,\\theta}(\\cO,T)\n:=\\ensuremath{\\mathbb{\\wso}}^0_{p,\\theta}(\\cO,T), \\quad\\quad\\quad \n\\bL^{[\\circ]}_{p,\\theta}(\\cO,T,\\ell_2)\n:=\\ensuremath{\\mathbb{\\wso}}^0_{p,\\theta}(\\cO,T;\\ell_2),\n\\]\nand\n\\[\n\\mathring{\\ensuremath{\\mathbb{\\wso}}}^1_{p,\\theta}(\\cO,T)\n:=\nL_p(\\Omega_T, \\ensuremath{\\mathcal{P}} ;\\mathring{\\ensuremath{K}}^{1}_{p,\\theta}(\\cO)).\n\\]\nMoreover, $\\cK^1_{p,\\theta,0}(\\domain,T)$ is defined the following way.\n\\begin{defn}\nLet $p \\geq 2$. We write $u\\in\\cK^1_{p,\\theta,0}(\\cO,T)$ if\n$u\\in\\mathring{\\ensuremath{\\mathbb{\\wso}}}^1_{p,\\theta-p}(\\cO,T)$\n and\nthere exist $f^0\\in \\bL^{[\\circ]}_{p,\\theta+p}(\\cO,T)$, $f^i\\in \\bL^{[\\circ]}_{p,\\theta}(\\cO,T)$, $i=1,2$, and\n $g\\in \\bL^{[\\circ]}_{p,\\theta}(\\cO,T;\\ell_2)$\nsuch that \n\\begin{equation*}\\label{eqn 28_2}\n du=(f^0+f^i_{x^i})\\, dt +g^k \\, dw^k_t,\\quad t\\in(0,T],\n\\end{equation*}\non $\\domain$ in the sense of distributions with $u(0,\\cdot)=0$; see Definition~\\ref{defn sol} accordingly. In this situation\nwe also write\n$$\n\\bD u:=f^0+f^i_{x^i}\\qquad\\text{and}\\qquad \\bS u :=g.\n$$\n\\end{defn}\n\n\n\n\nIn this article, Equation~\\eqref{eq:SHE:Intro} has the following meaning on $\\domain$.\n\n\\begin{defn}\nWe say that $u$ is a solution to Equation~\\eqref{eq:SHE:Intro} on $\\cO$ in the class $\\mathcal{K}^{1}_{p,\\theta,0}(\\cO,T)$ \nif\n$u\\in \\mathcal{K}^{1}_{p,\\theta,0}(\\cO,T)$ with\n\\[\n\\bD u = \\Delta u + f^0+f^i_{x^i}=f^0+(f^i+u_{x^i})_{x^i}\n\\qquad\n\\text{and}\n\\qquad\n\\bS u = g.\n\\]\n\\end{defn}\n\n\n\n\nBefore we look at Equation~\\eqref{eq:SHE:Intro} in detail, we first prove the following version of Lemma~\\ref{lem 4.5.1} for polygons.\nIt is a key ingredient in our existence and uniqueness proof below.\n\n\n\\begin{lemma}\n\\label{lem for gronwall}\nLet $p\\geq 2$ and $\\theta \\in \\bR$. Assume that $u\\in\\cK^1_{p,\\theta,0}(\\cO,T)$, such that $du=(f^0+f^i_{x^i})dt+g^kdw^k_t$ with\n\\[\nf^0\\in \\bL^{[\\circ]}_{p,\\theta+p}(\\cO,T), \n\\quad f^i \\in \\bL^{[\\circ]}_{p,\\theta}(\\cO,T),\\,i=1,2, \\quad\\text{and } g\\in \\bL^{[\\circ]}_{p,\\theta}(\\cO,T;\\ell_2).\n\\]\n Then $u\\in L_p(\\Omega; \\ensuremath{\\mathcal{C}}([0,T];L^{[\\circ]}_{p,\\theta}(\\cO)))$ and \n \\begin{align*}\n{\\mathbb E} \\sup_{t\\leq T}& \\|u(t,\\cdot)\\|^p_{L^{[\\circ]}_{p,\\theta}(\\cO)} \\\\\n&\\leq N \\Big(\\|u\\|^p_{\\ensuremath{\\mathbb{\\wso}}^{1}_{p,\\theta-p}(\\cO,T)}\n+\\|f^0\\|^p_{\\bL^{[\\circ]}_{p,\\theta+p}(\\cO, T)}+\\sum_i\\|f^i\\|^p_{\\bL^{[\\circ]}_{p,\\theta}(\\cO, T)}+\\|g\\|^p_{\\bL^{[\\circ]}_{p,\\theta}(\\cO, T; \\ell_2)} \\Big)\\\\\n&=:N \\,C(u,f^0,f^i,g, T),\n \\end{align*}\nwhere $N=N(d,p,\\theta,T)$ is a non-decreasing function of $T$.\nIn particular, for any $t\\leq T$,\n\\[\n\\|u\\|^p_{\\bL^{[\\circ]}_{p,\\theta}(\\cO,t)}\n\\leq \n\\int^t_0 {\\mathbb E}\\sup_{r\\leq s} \\|u(r)\\|^p_{L^{[\\circ]}_{p,\\theta}(\\cD)} ds\\leq N(d,p, \\theta, T) \\int^t_0 C(u,f^0,f^i,g, s) \\,ds.\n \\]\n \\end{lemma}\n\n\\begin{proof}\nWe combine Lemma \\ref {lem 4.5.1} (see also Remark~\\ref{remark cones}) and Lemma~\\ref{lem krylov} as follows.\n\nFix a sufficiently small $r>0$ such that $B_{3r}(v_j)$ contains only one vertex $v_j$ and intersects with only two edges for each $j\\leq M$. \nChoose a function $\\xi\\in \\ensuremath{\\mathcal{C}}^{\\infty}_c(\\bR^2)$ such that $0<\\xi(x)\\leq 1$ for $|x|<2r$, $\\xi(x)=1$ for $|x|c>0.\n \\end{equation}\n Note that by the choice of $\\xi_j$, $\\cD_j$, $j=0,1,\\ldots,M$, and $G$, for any $\\theta\\in\\bR$ \\color{black} and $v\\in \\mathring{K}\\color{black}^1_{p,\\theta-p}(\\cO)$, \n\\begin{equation}\n \\label{eqn 4.11.1}\n\\|\\xi_0 v\\|_{L^{[\\circ]}_{p,\\theta}(\\cO)} \n\\sim \n\\|\\xi_0 v\\|_{L_p(G)}, \\quad \\quad \n\\|\\xi_0v\\|_{K^1_{p,\\theta-p}(\\cO)}\\sim \\|\\xi_0v\\|_{H^1_{p,2-p}(G)},\n\\end{equation}\n\\begin{equation}\n \\label{eqn 4.11.2}\n\\|\\xi_j v\\|_{K^1_{p,\\theta-p}(\\cO)}=\\|\\xi_j v\\|_{K^1_{p,\\theta-p}(\\cD_j)} \\quad (j\\geq 1).\n \\end{equation}\nThe first and the third relation are trivial and hold actually for arbitrary measurable $v\\colon\\domain\\to \\bR$, provided the expressions make sense. The second one is due to \\eqref{eqn 4.10.12} and Hardy's inequality as\n \\begin{align*}\n\\|\\xi_0v\\|_{K^1_{p,\\theta-p}(\\cO)}\n&\\leq \nN (\\|\\xi_0v\\|_{L_p(G)}+\\sum_i\\|(\\xi_0v)_{x^i}\\|_{L_p(G)}) \\\\\n& \\leq N \\|\\xi_0 v\\|_{H^1_{p,2-p}(G)}\\leq N \\sum_i\\|(\\xi_0v)_{x^i}\\|_{L_p(G)} \n\\leq N \\|\\xi_0 v\\|_{K^1_{p,\\theta-p}(\\cO)}.\n \\end{align*}\n The three relations from~\\eqref{eqn 4.11.1} and~\\eqref{eqn 4.11.2} together imply, in particular, that \n\\[\n\\|v\\|_{L^{[\\circ]}_{p,\\theta}(\\cO)}\n\\sim \n\\sum_{j=0}^M \\|\\xi_j v\\|_{L^{[\\circ]}_{p,\\theta}(\\cO)} \n\\sim \n\\|\\xi_0v\\|_{L_p(G)}+ \\sum_{j=1}^M \\|\\xi_j v\\|_{L^{[\\circ]}_{p,\\theta}(\\cD_j)},\n\\]\n\\begin{eqnarray}\n \\label{eqn 4.11.4}\n\\|v\\|_{K^1_{p,\\theta-p}(\\cO)} \n\\sim \n\\sum_{j=0}^M \\|\\xi_j v\\|_{K^1_{p,\\theta-p}(\\cO)}\n\\sim\n\\|\\xi_0v\\|_{H^1_{p,2-p}(G)}\n+\n\\sum_{j=1}^M \\|\\xi_j v\\|_{K^1_{p,\\theta-p}(\\cD_j)}.\n\\end{eqnarray}\nAlso note that for any multi-index $\\alpha$,\n\\begin{equation}\n \\label{eqn 4.10.14}\n\\sum_{j=0}^M \\|v D^{\\alpha}\\xi_j\\|_{L^{[\\circ]}_{p,\\theta}(\\cO)}\n+ \n\\sum_{j=0}^M \\|v \\tilde{\\rho}D^{\\alpha}\\xi_j\\|_{L^{[\\circ]}_{p,\\theta}(\\cO)}\n\\leq N \n\\|v\\|_{L^{[\\circ]}_{p,\\theta}(\\cO)}.\n\\end{equation}\n\nUsing the preparations above, we can verify the assertion the following way. \nFor each $j\\in\\{1,2,\\ldots,M\\}$, $u^j:=\\xi_j u \\in \\cK^1_{p,\\theta,0}(\\cD_j,T)$ with\n\\begin{equation}\n \\label{eqn 4.10.11}\n du^j=\\big((\\xi_jf^i)_{x^i}+\\xi_jf^0-(\\xi_j)_{x^i}f^i\\big)\\,dt+ \\xi_jg^k\\, dw^k_t, \\quad t\\in(0,T],\n \\end{equation}\non $\\cD_j$ in the sense of distributions.\nThus, by Lemma~\\ref{lem 4.5.1} (see also Remark~\\ref{remark cones}) and (\\ref{eqn 4.10.10}),\n $ u^j \\in L_p(\\Omega; \\ensuremath{\\mathcal{C}}([0,T];L^{[\\circ]}_{p,\\theta}(\\cO)))$, and \n\\begin{align}\n{\\mathbb E} \\sup_{t\\leq T} \\|\\xi_j u\\|^p_{L^{[\\circ]}_{p,\\theta}(\\cO)}\n\\leq &N\\Big( \\|\\xi_j f^0\\|^p_{\\bL^{[\\circ]}_{p,\\theta+p}(\\cO,T)}+\\sum_i\\|\\xi_j f^i\\|^p_{\\bL^{[\\circ]}_{p,\\theta}(\\cO,T)}\\label{eqn 4.10.13}\\\\\n &+ \\|\\xi_ju\\|^p_{\\ensuremath{\\mathbb{\\wso}}^{1}_{p,\\theta-p}(\\cO,T)}+\\|(\\xi_j)_{x^i}\\tilde{\\rho}f^i \\|^p_{\\bL^{[\\circ]}_{p,\\theta}(\\cO,T)} +\n \\|\\xi_j g\\|^p_{\\bL^{[\\circ]}_{p,\\theta}(\\cO,T;\\ell_2)} \\Big). \\nonumber\n \\end{align}\nAlso, $u^0:=\\xi_0 u\\in \\frH^1_{p,2,0}(G,T)$ and \\eqref{eqn 4.10.11} holds with $j=0$. Thus, by Lemma~\\ref{lem krylov} and \\eqref{eqn 4.10.12}, $u^0\\in L_p(\\Omega; \\ensuremath{\\mathcal{C}}([0,T];L^{[\\circ]}_{p,\\theta}(\\cO)))$, and \\eqref{eqn 4.10.13} holds with $j=0$. Therefore, by summing up all these estimates and using above relations, we get the desired result. \n\\end{proof}\n\n\nOur main existence and uniqueness result for the stochastic heat equation on polygons reads as follows.\nRecall that in this section $\\kappa_0$ denotes the maximum over all interior angles of the polygon $\\cO$.\n\n\\begin{thm}[Existence and uniqueness\/polygons]\n \\label{thm polygon main}\nLet $p\\geq 2$ and assume that $\\theta\\in\\bR$ satisfies \\eqref{eq:range:vertex}. Then for any \n\\[\nf^0\\in \\bL^{[\\circ]}_{p,\\theta+p}(\\cO,T), \\quad\nf^i\\in \\bL^{[\\circ]}_{p,\\theta}(\\cO,T),\\,i=1,2,\\quad\\text{and}\\quad \ng \\in \\bL^{[\\circ]}_{p,\\theta}(\\cO,T;\\ell_2),\n\\]\nEquation~\\eqref{eq:SHE:Intro} on $\\cO$ \nhas a unique solution $u\\in\\cK^1_{p,\\theta,0}(\\cO,T)$. \nMoreover, \n\\begin{equation}\n \\label{eqn polygon main}\n\\|u\\|_{\\ensuremath{\\mathbb{\\wso}}^{1}_{p,\\theta-p}(\\cO,T)}\n\\leq \nN \\,\\sgrklam{\\|f^0\\|_{ \\bL^{[\\circ]}_{p,\\theta+p}(\\cO,T)}\n+\n\\sum_i\\|f^i\\|_ {\\bL^{[\\circ]}_{p,\\theta}(\\cO,T)}\n+\n\\|g\\|_{ \\bL^{[\\circ]}_{p,\\theta}(\\cO,T;\\ell_2)}},\n\\end{equation}\nwhere $N=N(p,\\theta,\\kappa_0,T)$.\n\\begin{proof}\n\\emph{Step 1.}\nWe first prove that~\\eqref{eqn polygon main} holds given that a solution $u\\in\\cK^1_{p,\\theta,0}(\\cO,T)$ already exists, by using corresponding results for the stochastic heat equation on angular domains and on $\\ensuremath{\\mathcal{C}}^1$ domains. This will, in particular, take care of the uniqueness. \n\n\n Let $r>0$, $\\xi_j$, $\\cD_j$, $j=1,\\ldots,M$, as well as $\\xi_0$ and $G$ be as in the proof of Lemma~\\ref{lem for gronwall}. \nA very similar reasoning as therein can be used to verify that $\\xi_0u\\in\\frH^{1}_{p,\\theta,0}(G,T)$, $\\xi_ju\\in\\cK^1_{p,\\theta,0}(\\cD_j,T)$ for $j\\geq 1$, and that for all $j\\in\\{0,1,\\ldots,M\\}$, \n\\begin{align*}\nd(\\xi_j u)\n&=\n\\grklam{\\Delta (\\xi_j u)+(-2u(\\xi_j)_{x^i}+\\xi_j f^i )_{x^i}\\\\\n&\\qquad\\qquad\\qquad\\qquad+u\\Delta \\xi_j-(\\xi_j)_{x^i}f^i+\\xi_jf^0}\\,dt+ \\xi_j g^k \\,dw^k_t, \\quad t\\in (0,T].\n\\end{align*}\nThus, by Theorem~\\ref{thm:ex:uni:2DCone} for $j\\geq 1$ and by Lemma~\\ref{lem 10} for $j=0$ (see also \\cite[Theorem~2.9]{Kim2004}), we obtain the estimate for \n$\\|\\xi_j u\\|^p_{\\ensuremath{\\mathbb{\\wso}}^{1}_{p,\\theta-p}(\\cO,t)}$ for each $t\\leq T$. Then summing up over all $j$ and using \\eqref{eqn 4.11.4} and \\eqref{eqn 4.10.14}, yields that for each $t\\leq T$,\n\\begin{align}\n\\|u\\|_{\\ensuremath{\\mathbb{\\wso}}^{1}_{p,\\theta-p}(\\cO,t)}\n&\\leq N \\sgrklam{\n\\|u\\|^p_{\\bL^{[\\circ]}_{p,\\theta}(\\cO,t)}\n+\\label{2018-7}\n\\|f^0\\|^p_{\\bL^{[\\circ]}_{p,\\theta+p}(\\cO,T)}\\\\\n&\\qquad\\qquad\\qquad\\qquad +\n\\sum_i\\|f^i\\|^p_{\\bL^{[\\circ]}_{p,\\theta}(\\cO,T)}\n+\n\\|g\\|^p_{\\bL^{[\\circ]}_{p,\\theta}(\\cO,T;\\ell_2)} }. \\nonumber \n\\end{align}\nRecall that\n$$\ndu=(f^0+(f^i+u_{x^i})_{x^i})\\,dt+g^k\\,dw^k_t, \\quad t\\leq T,\n$$\nand $\\|u_x\\|_{\\bL^{[\\circ]}_{p,\\theta}(\\cO,s)}\\leq N \\|u\\|_{\\ensuremath{\\mathbb{\\wso}}^1_{p,\\theta-p}(\\cO,s)}$. \nThus, by Lemma~\\ref{lem for gronwall} and \\eqref{2018-7}, for each $t\\leq T$,\n\\begin{align*} \n\\|u\\|^p_{\\ensuremath{\\mathbb{\\wso}}^{1}_{p,\\theta-p}(\\cO,t)} \n&\\leq N \\int^t_0 \\|u\\|^p_{\\ensuremath{\\mathbb{\\wso}}^{1}_{p,\\theta-p}(\\cO,s)} \\,ds\\\\\n&\\qquad + N\\sgrklam{\\|f^0\\|^p_{\\bL^{[\\circ]}_{p,\\theta+p}(\\cO,T)}+ \\sum_i\\|f^i\\|^p_{\\bL^{[\\circ]}_{p,\\theta}(\\cO,T)}\n + \\|g\\|^p_{\\bL^{[\\circ]}_{p,\\theta}(\\cO,T;\\ell_2)}}.\n\\end{align*}\nHence the desired estimate follows by Gronwall's inequality.\n\n\\smallskip\n \n\\noindent\\emph{Step 2.} We prove existence as follows.\n Due to Lemma~\\ref{lem for gronwall} \nand the a-priori estimate obtained in Step~1, we may assume $f^0$, $f^i$, $i=1,2$, and $g$ are very nice in the sense that they vanish near the boundary and \n\\[\nf^i, f^i_{x^i}, f^0\\in L_2(\\Omega_T,\\cP; L_2(\\cO)), \\quad \\text{and}\\quad g \\in L_2(\\Omega_T,\\cP; L_2(\\cO;\\ell_2)).\n\\]\nThen, by classical results (see, for instance, \\cite{Roz1990} or \\cite[Theorem~2.12]{Kim2014}), there exists a unique solution $u$ in $\\frH^1_{2,2,0}(\\cO,T)$, which satisfies, in particular, \n\\begin{equation}\n \\label{eqn 4.11.7}\n\\rho^{-1}u, u_{x^i} \\in L_2(\\Omega_T,\\cP;L_2(\\cO)),\\;i=1,2, \\;\\text{ and }\\; \\sup_x |u|\\leq N \\|u_{x}\\|_{L_2(\\cO)}.\n \\end{equation}\nNote that for each $j \\geq 1$,\n\\begin{equation}\n\\label{eqn 4.11.10}\nd(\\xi_j u)=\\grklam{\\Delta (\\xi_j u)+f^{j,i}_{x^i}+f^{j,0} }\\,dt+ \\xi_j g^k \\,dw^k_t, \\quad t\\in(0,T],\n\\end{equation}\non $\\cD_j$, where, due to \\eqref{eqn 4.11.7} and the fact that $(\\xi_j)_{x^i}=0$ near the vertex $v_j$,\n\\begin{equation}\n\\label{fji}\nf^{j,i}:=-2(\\xi_j)_{x^i} u+\\xi_jf^i \\in \\bL^{[\\circ]}_{p,\\theta}(\\cD_j,T) \\cap \\bL^{[\\circ]}_{2,2}(\\cD_j,T),\n\\end{equation}\n\\begin{equation}\n\\label{fj0}\nf^{j,0}:=u\\Delta \\xi_j+f^0\\xi_j + \\sum_i (\\xi_j)_{x_i}f^i \\in \\bL^{[\\circ]}_{p,\\theta+p}(\\cD_j,T) \\cap \\bL^{[\\circ]}_{2,2+2}(\\cD_j,T), \n\\end{equation}\nand \n\\[\n\\xi_j g \\in \\bL^{[\\circ]}_{p,\\theta}(\\cD_j,T;\\ell_2) \\cap \\bL^{[\\circ]}_{2,2}(\\cD_j,T;\\ell_2).\n\\]\nSince $\\tilde{\\rho}(x) \\geq \\rho(x)$, it follows that for each for $j\\geq 1$ we have $\\xi_ju\\in \\cK^1_{2,2,0}(\\cO,T)$. Thus, by Lemma \\ref{lem for uniqueness}, we conclude $\\xi_ju\\in \\cK^1_{p,\\theta,0}(\\cO,T)$ if $j\\geq 1$. Similar arguments based on Lemma~\n\\ref{lem for uniqueness2} yield that $\\xi_0 u\\in \\frH^1_{p,2,0}(G,T)$. Therefore, $\\xi_0 u\\in \\cK^1_{p,\\theta,0}(\\cO,T)$ (see~\\eqref{eqn 4.11.1}), and consequently \n$u\\in \\cK^1_{p,\\theta,0}(\\cO,T)$.\n\\end{proof}\n\\end{thm}\n\n\n\n\\begin{remark}\n\\label{remark 8.24}\n Note that even if we were to consider Equation~\\eqref{eq:SHE:Intro} on $\\domain$ with $f^i=0$, $i=1,2$, our proof strategy for Theorem~\\ref{thm polygon main} (and Theorem~\\ref{thm_polygons_1} below) requires that we are able to handle the localized equation on $\\cD_j$ with forcing term $((\\xi_j)_{x^i}u)_{x^i}$, which means that we have to be able to treat Equation~\\eqref{eq:SHE:Intro} on angular domains with $f^i\\neq 0$. \nThis is why we need the extension of \\cite[Theorem~3.7]{CioKimLee+2018} presented in Theorem~\\ref{thm:ex:uni:2DCone} even for the proof of Theorem~\\ref{thm polygon main} with $f^i=0$, $i=1,2$.\n\\end{remark}\n\n\n\nWe conclude with our main higher order regularity result for the stochastic heat equation on polygons. \n\n\n\\begin{thm}[Higher order regularity\/polygons]\\label{thm_polygons_1}\n Given the setting of Theorem~\\ref{thm polygon main}, let $u$ be the unique solution in the class $\\cK^{1}_{p,\\theta,0}(\\cO,T)$ to Equation~\\eqref{eq:SHE:Intro} on $\\cO$. \nAssume that \n\\begin{align*}\nC(m,\\theta, f^i,f^0,g)\n&:=\n{\\mathbb E} \\int^T_0 \\int_{\\cO} \\ssgrklam{\\sum_{|\\alpha|\\leq (m-1)\\vee 0\\color{black}} |\\rho^{\\abs{\\alpha}+1}D^{\\alpha}f^0|^p+ \\sum_i\\sum_{|\\alpha|\\leq m} |\\rho^{|\\alpha|}D^{\\alpha}f^i|^p\\\\\n&\\qquad\\qquad\\qquad\\qquad+|\\tilde{\\rho}f^0|^p+\n\\sum_{\\abs{\\alpha}\\leq m} |\\rho^{\\abs{\\alpha}}D^\\alpha g|_{\\ell_2}^p} \\tilde{\\rho}^{\\theta-2}\\, dx\\,dt <\\infty,\n\\end{align*}\n for some $m\\in\\{0,1,2,\\ldots\\}$.\nThen\n\\begin{equation}\n \\label{eqn final}\n{\\mathbb E}\\int_0^T \\sum_{\\abs{\\alpha}\\leq m+1}\\int_{\\cO} \\Abs{\\rho^{\\abs{\\alpha}-1}D^\\alpha u}^p \\tilde{\\rho}^{\\theta-2}\\,dx\\,dt\\leq N\\, C(m,\\theta, f^i,f^0,g),\n\\end{equation}\nwhere $N=N(p,\\theta,\\kappa_0,m, T)$.\n\\end{thm}\n\n\n\n\\begin{proof}\nWe prove the statement by induction over $m$. As in the proof of the results above, we use a partition of unity and apply corresponding results for the stochastic heat equation on angular domains (Corollary~\\ref{high}) and on $\\ensuremath{\\mathcal{C}}^1$ domains (\\cite[Theorem~2.9]{Kim2004}) to estimate the solutions of the localized equations.\n\nLet $r>0$, $\\xi_j$, $\\cD_j$, $j=1,\\ldots,M$, as well as $\\xi_0$ and $G$ be as in the proof of Lemma~\\ref{lem for gronwall}.\nIn addition, assume that $G\\subset\\domain$ is chosen in such a way that\n\\[\n\\domain\\setminus\\bigcup_{j}B_{2r\/3}(v_j) \\subseteq G\\subseteq \\domain\\setminus\\bigcup_{j}B_{r\/3}(v_j).\n\\]\nAs a consequence, \n\\begin{equation}\\label{eq:equiv:dist}\n\\rho_G\\sim\\rho_\\cO\\quad \\text{and}\\quad \\tilde\\rho\\sim 1\\quad \\text{on}\\quad\\textup{supp}\\,\\xi_0\\cap\\domain.\n\\end{equation}\n \n\\smallskip\n \n\\noindent\\emph{Step 1. The base case.} Let $m=0$. \nNote that in this case, the only difference in Estimate~\\eqref{eqn final} compared to~\\eqref{eqn polygon main} is the weight we put on $u$ on the left hand side of the inequality: $\\rho^{-p}\\tilde\\rho^{\\theta-2}$ in~\\eqref{eqn final} instead of the smaller $\\tilde\\rho^{\\theta-p-2}$ from~\\eqref{eqn polygon main}.\nBut to obtain this sharper estimate we argue in a very similar fashion as in the proof of the latter with two changes: \nWe use~Corollary~\\ref{high} instead of Theorem~\\ref{thm:ex:uni:2DCone} to estimate the solution in the vicinity of vertexes and we use the slightly modified choice of $G$ and~\\eqref{eq:equiv:dist} to replace $\\rho_G$ by $\\rho_\\domain$ after applying~\\cite[Theorem~2.9]{Kim2004} to estimate the solution away from the vertexes. \nIn detail, we argue as follows: The same reasoning as in the proof of Theorem~\\ref{thm polygon main} shows that $\\xi_0 u\\in \\frH^1_{p,\\theta,0}(G,T)$ and $\\xi_j u\\in\\cK^1_{p,\\theta,0}(\\cD_j,T)$ for $j\\geq 1$ satisfy~\\eqref{eqn 4.11.10} on $G$ and on $\\cD_j$, $j\\geq 1$, respectively. \nIn particular, if $1\\leq j\\leq M$, then \nby Corollary~\\ref{high} (see also Remark~\\ref{remark cones}), Estimate~\\eqref{eqn final} holds with $\\xi_j u$ and $C(0, \\theta,f^{j,i}, f^{j,0},\\xi_jg)$ in place of $u$ and $C(0,\\theta,f^i,f^0, g)$, respectively. Here $f^{j,i}$ and $f^{j,0}$ are taken from \\eqref{fji} and \\eqref{fj0}.\nMoreover, by the corresponding result on $\\ensuremath{\\mathcal{C}}^1$ domains (see \\cite[Theorem~2.9]{Kim2004}) and \\eqref{eq:equiv:dist}, Estimate~\\eqref{eqn final} also holds for $\\xi_0u$ and $C(0,\\theta, f^{0,i}, f^{0,0},\\xi_0g)$ in place of $u$ and $C(0,\\theta,f^i,f^0, g)$, respectively. \n Summing up all these estimates and using the second relationship in~\\eqref{eq:equiv:dist} yields \n\\begin{align*}\n{\\mathbb E}\\int_0^T \\sum_{\\abs{\\alpha}\\leq 1} \\int_{\\cO} \\Abs{\\rho^{\\abs{\\alpha}-1}D^\\alpha u}^p \\tilde{\\rho}^{\\theta-2}\\,dx\\,dt\n&\\leq N \\sum_{j=0}^{M} C(0,\\theta, f^{j,i},f^{j,0},\\xi_j g)\\\\\n&\\leq N \\,\\|u\\|^p_{\\bL^{[\\circ]}_{p,\\theta-p}(\\cO)}+ N C(0,\\theta,f^i,f^0,g) \\\\\n&\\leq N\\, C(0,\\theta,f^i,f^0,g);\n\\end{align*}\nthe last inequality above is due to \\eqref{eqn polygon main}. The base case is proved.\n\n\\smallskip\n \n\\noindent\\emph{Step 2. The induction step.} Suppose that \\eqref{eqn final} holds for some $m\\in\\{0,1,\\ldots\\}$ and $C(m+1,\\theta,f^i,f^0,g)<\\infty$. Then, by assumption,\n\\begin{equation}\\label{eqn 4.13.2.a}\n{\\mathbb E}\\int_0^T \\sum_{\\abs{\\alpha}\\leq m+1}\\int_{\\cO} \\Abs{\\rho^{\\abs{\\alpha}-1}D^\\alpha u}^p \\tilde{\\rho}^{\\theta-2}\\,dx\\,dt\\leq N C(m,\\theta, f^i,f^0,g).\n\\end{equation}\nUsing \\eqref{eqn 4.13.2.a}, one can easily check that\n $$\n \\sum_{j=0}^{M} C(m+1,\\theta,f^{j,i}, f^{j,0},\\xi_j g) \\leq N\\, C(m+1,\\theta,f^i,f^0,g).\n $$ \nTherefore, appropriate applications of Corollary~\\ref{high} (see also Remark~\\ref{remark cones}) and \\cite[Theorem~2.9]{Kim2004} yield suitable estimates of $\\sum_{|\\alpha|\\leq m+2} {\\mathbb E} \\int^T_0 |\\rho^{|\\alpha|-1}D^{\\alpha}(\\xi_j u)|^p \\tilde{\\rho}^{\\theta-2}dxdt$ for $j=0$ and $1\\leq j\\leq M$, respectively, which, summed up, yield\n\\begin{align*}\n{\\mathbb E}\\int_0^T \\sum_{\\abs{\\alpha}\\leq m+2} \\int_{\\cO}& \\Abs{\\rho^{\\abs{\\alpha}-1}D^\\alpha u}^p \\tilde{\\rho}^{\\theta-2}\\,dx\\,dt\\\\\n& \\leq N \\sum_{j=0}^{M}\n C(m+1,\\theta,f^{j,i},f^{j,0},\\xi_j g) \\leq N\\, C(m+1,\\theta,f^i,f^0,g). \n\\end{align*}\nThus the induction goes through and the theorem is proved.\n\\end{proof}\n\n\n\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]{%\n \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nFix a non-archimedean local field ${\\mathrm{k}}$ of characteristic zero, and\na continuous involution $\\tau$ on it. Denote by ${\\mathrm{k}}_0$ the fixed\npoints of $\\tau$. Then either ${\\mathrm{k}}={\\mathrm{k}}_0$ or ${\\mathrm{k}}$ is a quadratic\nextension of ${\\mathrm{k}}_0$. Let $\\epsilon=\\pm 1$ and let $E$ be an\n$\\epsilon$-hermitian space, namely it is a finite dimensional\n${\\mathrm{k}}$-vector space, equipped with a non-degenerate ${\\mathrm{k}}_0$-bilinear\nmap\n\\[\n \\langle\\,,\\,\\rangle_E:E\\times E\\rightarrow {\\mathrm{k}}\n\\]\nsatisfying\n\\[\n \\langle u,v\\rangle_E=\\epsilon\\langle v,u\\rangle_E^\\tau, \\quad \\langle au,v\\rangle_E=a\\langle u,\n v\\rangle_E,\\quad a\\in A,\\, u,v\\in E.\n\\]\nWrite $\\epsilon'=-\\epsilon$, and let $(E',\\langle\\,,\\,\\rangle_{E'})$ be an\n$\\epsilon'$-hermitian space. Then\n\\[\n \\mathbf{E}:=E\\otimes_{\\mathrm{k}} E'\n\\]\nis a ${\\mathrm{k}}_0$-symplectic space under the form\n\\[\n \\langle u\\otimes u', v\\otimes v'\\rangle_\\mathbf{E}:={\\mathrm{tr}}_{{\\mathrm{k}}\/{\\mathrm{k}}_0}(\\langle u, v\\rangle_E \\, \\langle\n u',v'\\rangle_{E'}).\n\\]\n\nDenote by\n\\begin{equation}\\label{meta}\n 1\\rightarrow \\{\\pm 1\\}\\rightarrow\n \\widetilde{{\\mathrm{Sp}}}(\\mathbf{E})\\rightarrow\n {\\mathrm{Sp}}(\\mathbf{E})\\rightarrow 1\n\\end{equation}\nthe metaplectic cover of the symplectic group ${\\mathrm{Sp}}(\\mathbf{E})$. Denote by\n\\[\n \\H:=\\mathbf{E}\\times {\\mathrm{k}}_0\n\\]\nthe Heisenberg group associated to $\\mathbf{E}$, whose multiplication is\ngiven by\n\\[\n (u,t)(u',t'):=(u+u', t+t'+\\langle u,u'\\rangle_\\mathbf{E}).\n\\]\nThe group ${\\mathrm{Sp}}(\\mathbf{E})$ acts on $\\H$ as automorphisms by\n\\begin{equation}\\label{actsp0}\n g.(u,t):=(gu,t).\n\\end{equation}\nIt induces an action of $\\widetilde{{\\mathrm{Sp}}}(\\mathbf{E})$ on $\\H$, and further\ndefines a semidirect product $\\widetilde{{\\mathrm{Sp}}}(\\mathbf{E})\\ltimes \\H$.\n\nFix a non-trivial character $\\psi$ of ${\\mathrm{k}}$, and denote by\n${\\omega}_\\psi$ the corresponding smooth oscillator representation\nof $\\widetilde{{\\mathrm{Sp}}}(\\mathbf{E})\\ltimes \\H$. Up to isomorphism, this is the\nonly genuine smooth representation which, as a representation of\n$\\H$, is irreducible and has central character $\\psi$. Recall that\nin general, if $H$ is a group together with an embedding of $\\{\\pm\n1\\}$ in its center, a representation of $H$ is called genuine if the\nelement $-1\\in H$ acts via the scalar multiplication by $-1$.\n\n\nDenote by $G$ the group of all ${\\mathrm{k}}$-linear automorphisms of $E$\nwhich preserve the form $\\langle\\,,\\,\\rangle_E$. It is thus an orthogonal\ngroup, a symplectic group or a unitary group. The group $G$ is\nobviously mapped into ${\\mathrm{Sp}}(\\mathbf{E})$. Define the fiber product\n\\[\n \\widetilde{G}:=\\widetilde{{\\mathrm{Sp}}}(\\mathbf{E})\\times_{{\\mathrm{Sp}}(\\mathbf{E})} G,\n\\]\nwhich is a double cover of $G$. Similarly, we define $G'$ and\n$\\widetilde G'$. As usual, the product group $\\widetilde{G}\\times\n\\tilde{G'}$ is mapped into $\\widetilde{{\\mathrm{Sp}}}(\\mathbf{E})$. {\\vspace{0.2in}}\n\n\nThe goal of this paper is to prove the following theorem, which is\nusually called the multiplicity preservation for theta\ncorrespondences, and is also called the Multiplicity One Conjecture\nby Rallis in \\cite{Ra84}. It is complementary to the famous Local\nHowe Duality Conjecture.\n\\begin{introtheorem}\\label{theorem}\nFor every genuine irreducible admissible smooth representation $\\pi$\nof $\\widetilde{G}$, and $\\pi'$ of $\\widetilde{G}'$, one has that\n\\[\n \\dim {\\mathrm{Hom}}_{\\widetilde{G}\\times \\widetilde{G}'}(\\omega_\\psi,\n \\pi\\otimes \\pi')\\leq 1.\n\\]\n\\end{introtheorem}\n\nWhen the residue characteristic of ${\\mathrm{k}}$ is odd, Theorem A is proved\nby Waldspurger in \\cite{Wa90}. The archimedean analog of Theorem A\nis proved by Howe in \\cite{Ho89}.\n\n\\section{A geometric result}\nWe continue with the notation of the Introduction. Following\n\\cite[Proposition 4.I.2]{MVW87}, we extend $G$ to a larger group\n$\\breve{G}$, which contains $G$ as a subgroup of index two, and\nconsists pairs $(g,\\delta)\\in{\\mathrm{GL}}_{{\\mathrm{k}}_0}(E)\\times \\{\\pm 1\\}$ such\nthat either\n\\[\n \\delta=1 \\quad\\textrm{and}\\quad g\\in G,\n\\]\nor\n\\[\n\\label{dutilde}\n \\left\\{\n \\begin{array}{ll}\n \\delta=-1,&\\medskip\\\\\n g(au)=a^\\tau g(u),\\quad & a\\in {\\mathrm{k}},\\, u\\in E,\\quad \\textrm{ and}\\medskip\\\\\n \\langle gu,gv\\rangle_E=\\langle v,u\\rangle_E,\\quad & u,v\\in E.\n \\end{array}\n \\right.\n\\]\nSimilarly, we define a group $\\breve{G'}$ and a group\n$\\breve{{\\mathrm{Sp}}}(\\mathbf{E})$, which extend $G'$ and ${\\mathrm{Sp}}(\\mathbf{E})$, respectively.\n\nIn general, if a group $\\breve H$ is equipped with a subgroup $H$ of\nindex two, we will associate on it the nontrivial quadratic\ncharacter which is trivial on $H$. We use $\\chi_H$ to indicate this\ncharacter.\n\nDenote the fiber product\n\\[\n \\breve{\\mathbf{G}}:=\\breve{G}\\times_{\\{\\pm 1\\}} \\breve{G'}=\\{(g,g',\\delta)\\mid (g,\\delta)\\in \\breve{G},\\,(g',\\delta)\\in \\breve{G'}\\},\n\\]\nwhich contains\n\\[\n \\mathbf{G}:=G\\times G'\n\\]\nas a subgroup of index two. Define a group homomorphism\n\\begin{equation}\\label{xi}\n \\begin{array}{rcl}\n \\xi:\\breve{\\mathbf{G}}&\\rightarrow &\\breve{{\\mathrm{Sp}}}(\\mathbf{E}),\\smallskip\\\\\n (g,g',\\delta)&\\mapsto& (g\\otimes g',\\delta).\\\\\n \\end{array}\n\\end{equation}\nLet $\\breve{{\\mathrm{Sp}}}(\\mathbf{E})$ act on the Heisenberg group $\\H$ as group\nautomorphisms by\n\\begin{equation}\\label{actsp}\n (g,\\delta).(u,t):=(gu, \\delta t),\n\\end{equation}\nwhich extends the action (\\ref{actsp0}). By using the homomorphism\n$\\xi$, this induces an action of $\\breve{\\mathbf{G}}$ on $\\H$, and further\ndefines a semidirect product\n\\[\n \\breve{\\mathbf{J}}:=\\breve{\\mathbf{G}}\\ltimes \\H,\n\\]\nwhich contains\n\\[\n \\mathbf{J}:=\\mathbf{G}\\ltimes \\H\n\\]\nas a subgroup of index two.\n\nLet the group\n\\begin{equation}\\label{semid}\n \\{\\pm 1\\}\\ltimes (\\breve{\\mathbf{G}}\\times \\breve{\\mathbf{G}})\n\\end{equation}\nact on $\\breve{\\mathbf{J}}$ by\n\\begin{equation}\\label{actionall}\n (\\delta, \\breve{\\mathbf g}_1, \\breve{\\mathbf g}_2). \\breve{\\mathbf j}:=(\\breve{\\mathbf g}_1\\,\n \\breve{\\mathbf j}\\, \\breve{\\mathbf g}_2^{-1})^\\delta,\n\\end{equation}\nwhere the semidirect product in (\\ref{semid}) is defined by the\naction\n\\[\n -1.(\\breve{\\mathbf g}_1, \\breve{\\mathbf g}_2):=(\\breve{\\mathbf g}_2, \\breve{\\mathbf g}_1).\n\\]\nThe fibre product\n\\[\n \\{\\pm 1\\}\\ltimes_{\\{\\pm 1\\}} (\\breve{\\mathbf{G}}\\times_{\\{\\pm 1\\}}\n \\breve{\\mathbf{G}})=\\{(\\delta,\\breve{\\mathbf g}_1, \\breve{\\mathbf\n g}_2)\\mid \\chi_\\mathbf{G}(\\breve{\\mathbf g}_1)=\\chi_\\mathbf{G}(\\breve{\\mathbf\n g}_2)=\\delta\\}\n\\]\nis a subgroup of (\\ref{semid}). It contains $\\mathbf{G}\\times \\mathbf{G}$ as a\nsubgroup of index two, and stabilizes $\\mathbf{J}$ under the action\n(\\ref{actionall}).\n\nWe prove the following proposition in the remaining of this section.\n\n\\begin{prpp}\\label{orbite1}\nEvery $\\mathbf{G}\\times \\mathbf{G}$-orbit in $\\mathbf{J}$ is stable under the group $\\{\\pm\n1\\}\\ltimes_{\\{\\pm 1\\}}(\\breve{\\mathbf{G}}\\times_{\\{\\pm 1\\}}\\breve{\\mathbf{G}})$.\n\\end{prpp}\n\n{\\vspace{0.2in}} Let $\\breve{\\mathbf{G}}$ act ${\\mathrm{k}}_0$-linearly on $\\mathbf{E}$ by\n\\begin{equation}\\label{acte}\n (g,g',\\delta).u\\otimes u':=\\delta gu\\otimes g'u'.\n\\end{equation}\n\n\\begin{lemp}\\label{orbite2}\nEvery $\\mathbf{G}$-orbit in $\\mathbf{E}$ is $\\breve{\\mathbf{G}}$-stable.\n\\end{lemp}\n\nWe first prove\n\n\\begin{lemp}\\label{orbite3}\nLemma \\ref{orbite2} implies Proposition \\ref{orbite1}.\n\\end{lemp}\n\\begin{proof}\nNote that every $\\mathbf{G}\\times \\mathbf{G}$-orbit in $\\mathbf{J}$ intersect the subgroup\n$\\H$, and the subgroup\n\\[\n \\{\\pm 1\\}\\times_{\\{\\pm 1\\}}(\\Delta(\\breve{\\mathbf{G}})) \\quad\\textrm{ of }\\quad \\{\\pm 1\\}\\ltimes_{\\{\\pm 1\\}}(\\breve{\\mathbf{G}}\\times_{\\{\\pm\n1\\}}\\breve{\\mathbf{G}}) \\] stabilizes $\\H$, where ``$\\Delta$\" stands for the\ndiagonal group. Therefore in order to prove Proposition\n\\ref{orbite1}, it suffices to show that every $\\Delta(\\mathbf{G})$-orbit in\n$\\H$ is $\\{\\pm 1\\}\\times_{\\{\\pm 1\\}}(\\Delta(\\breve{\\mathbf{G}}))$-stable.\nIdentify $\\{\\pm 1\\}\\times_{\\{\\pm 1\\}}(\\Delta(\\breve{\\mathbf{G}}))$ with\n$\\breve{\\mathbf{G}}$. Then as a $\\breve{\\mathbf{G}}$-space,\n\\[\n \\H=\\mathbf{E}\\times {\\mathrm{k}},\n\\]\nwhere $\\mathbf{E}$ carries the action (\\ref{acte}), and ${\\mathrm{k}}$ carries the\ntrivial $\\breve{\\mathbf{G}}$-action. This finishes the proof.\n\\end{proof}\n\n\nLet $\\breve{\\mathbf{G}}$ act ${\\mathrm{k}}_0$-linearly on\n\\[\n \\mathbf{E}':={\\mathrm{Hom}}_{\\mathrm{k}}(E,E')\n\\]\nby\n\\[\n ((g,g',\\delta).\\phi)(u):=\\delta \\,g'(\\phi(g^\\tau u)),\n\\]\nwhere\n\\[\n g^\\tau:=\\left\\{\n \\begin{array}{ll}\n g^{-1}, &\\quad\n \\textrm{if } \\delta=1,\\medskip\\\\\n \\epsilon g^{-1},& \\quad\n \\textrm{if } \\delta=-1.\\\\\n \\end{array}\n \\right.\n\\]\nThen one checks that the ${\\mathrm{k}}_0$-linear isomorphism\n\\[\n \\begin{array}{rcl}\n \\mathbf{E}&\\rightarrow &\\mathbf{E}',\\\\\n u\\otimes u'&\\mapsto&(v\\mapsto \\langle v,u\\rangle_E u')\n \\end{array}\n\\]\nis $\\breve{\\mathbf{G}}$-intertwining. Therefore Lemma \\ref{orbite2} is\nequivalent to the following\n\\begin{lemp}\\label{orbite4}\nEvery $\\mathbf{G}$-orbit in $\\mathbf{E}'$ is $\\breve{\\mathbf{G}}$-stable.\n\\end{lemp}\n\n\nDenote by\n\\[\n \\mathfrak g:=\\{x\\in {\\mathrm{End}}_{\\mathrm{k}}(E)\\mid \\langle xu,v\\rangle_E+\\langle u,xv\\rangle_E=0\\}\n\\]\nthe Lie algebra of $G$, and put\n\\[\n \\tilde{\\mathfrak g}:=\\{(x,F)\\mid x\\in \\mathfrak g, F\\,\\textrm{ is a ${\\mathrm{k}}$-subspace of }E,\n \\,x|_F=0\\}.\n\\]\nLet $\\breve G$ act on $\\tilde \\mathfrak g$ by\n\\[\n (g,\\delta). (x,F):=(\\delta gxg^{-1}, gF).\n\\]\nThe action of $\\breve{\\mathbf{G}}$ on $\\mathbf{E}'$ induces an action of\n\\[\n \\breve{G}=\\breve{\\mathbf{G}}\/G'\n\\]\non the quotient space $G'\\backslash\\mathbf{E}'$.\n\n\n\n\\begin{lemp}\\label{orbite5}\nThere is a $\\breve G$-intertwining embedding from $G'\\backslash\\mathbf{E}'$\ninto $\\tilde \\mathfrak g$.\n\\end{lemp}\n\\begin{proof}\nRecall that the map\n\\[\n x\\mapsto\\langle x\\,\\cdot,\\,\\cdot\\rangle_E\n\\]\nestablishes a ${\\mathrm{k}}_0$-linear isomorphism form $\\mathfrak g$ onto the space of\n$\\epsilon'$-hermitian forms on the ${\\mathrm{k}}$-vector space $E$. Define a\nmap\n\\[\n \\begin{array}{rcl}\n \\Xi: \\mathbf{E}'={\\mathrm{Hom}}_{\\mathrm{k}}(E,E')&\\rightarrow &\\tilde \\mathfrak g,\\\\\n \\phi&\\mapsto &(x,F),\n \\end{array}\n\\]\nwhere $F$ is the kernel of $\\phi$, and $x$ is specified by the\nformula\n\\[\n \\langle \\phi(u),\\phi(v)\\rangle_{E'}=\\langle xu,v\\rangle_E,\\quad u,v\\in E.\n\\]\nUse Witt's Theorem, one finds that two elements of $\\mathbf{E}'$ stay in the\nsame $G'$-orbit precisely when they have the same image under the\nmap $\\Xi$. Therefore $\\Xi$ reduces to an embedding\n\\[\n G'\\backslash\\mathbf{E}'\\hookrightarrow\\tilde \\mathfrak g,\n\\]\nwhich is checked to be $\\breve G$-intertwining.\n\\end{proof}\n\n\nThe following lemma is stated in \\cite[Proposition 4.I.2]{MVW87}. We\nomit its proof.\n\n\\begin{lemp}\\label{geometry}\nFor every $(x,F)\\in \\tilde \\mathfrak g$, there is an element $(g,-1)\\in\n\\breve{G}$ such that \\[\n gxg^{-1}=-x\\quad\\textrm{ and }\\quad gF=F.\n\\]\n\\end{lemp}\n\nIn other words, every element of $\\tilde \\mathfrak g$ is fixed by an element\nof $\\breve G\\setminus G$. Therefore every $G$-orbit in $\\tilde \\mathfrak g$\nis $\\breve G$-stable. Now Lemma \\ref{orbite5} implies that every\n$G$-orbit in $G'\\backslash \\mathbf{E}'$ is $\\breve G$-stable, or\nequivalently, every $\\mathbf{G}$-orbit in $\\mathbf{E}'$ is $\\breve \\mathbf{G}$-stable. This\nproves Lemma \\ref{orbite4}, and the proof of Proposition\n\\ref{orbite1} is now complete.\n\n\n\\section{Proof of Theorem \\ref{theorem}}\n\n\nWe first recall the notions of distributions and generalized\nfunctions on a t.d. group, i.e., a topological group whose\nunderlying topological space is Hausdorff, secondly countable,\nlocally compact and totally disconnected. Let $H$ be a t.d. group. A\ndistribution on $H$ is defined to be a linear functional on\n$\\textit{C}^\\infty_0(H)$, the space of compactly supported, locally\nconstant (complex valued) functions on $H$. Denote by\n$\\textit{D}^\\infty_0(H)$ the space of compactly supported distributions on\n$H$ which are locally scalar multiples of a fixed haar measure. A\ngeneralized function on $H$ is defined to be a linear functional on\n$\\textit{D}^\\infty_0(H)$.\n\n\n\nRecall the following version of Gelfand-Kazhdan criteria.\n\n\\begin{lem}\\label{gelfand}\nLet $S$ be a closed subgroup of a t.d. group $H$, and let $\\sigma$\nbe a continuous anti-automorphism of $H$. Assume that every\nbi-$S$-invariant generalized function on $H$ is $\\sigma$-invariant.\nThen for every irreducible admissible smooth representations $\\pi$\nof $H$, one has that\n\\begin{equation*}\n \\dim {\\mathrm{Hom}}_{S}(\\pi, \\mathbb{C}) \\,\\cdot\\, \\dim {\\mathrm{Hom}}_{S}\n (\\pi^{\\vee},\\mathbb{C})\\leq 1.\n\\end{equation*}\n\\end{lem}\nHere and henceforth, we use ``$^{\\vee}$\" to indicate the contragredient\nof an admissible smooth representation. Lemma \\ref{gelfand} is\nproved in a more general form in \\cite[Theorem 2.2]{SZ} for real\nreductive groups. The same proof works here and we omit the details.\n\n{\\vspace{0.2in}}\n\nNow we continue with the notation of the last section.\n\n\\begin{leml}\\label{invgeneralized}\nIf a generalized function on $\\mathbf{J}$ is $\\mathbf{G}\\times \\mathbf{G}$ invariant, then\nit is also invariant under the group $\\{\\pm 1\\}\\ltimes_{\\{\\pm\n1\\}}(\\breve{\\mathbf{G}}\\times_{\\{\\pm 1\\}}\\breve{\\mathbf{G}})$.\n\\end{leml}\n\\begin{proof}\nNote that the t.d. group $\\breve{\\mathbf{J}}$ is unimodular. Therefore we\nmay replace ``generalized function\" by ``distribution\" in the proof\nof the lemma. Then by \\cite[Theorem 6.9 and Theorem 6.15 A]{BZ76},\nthe lemma is implied by Proposition \\ref{orbite1}.\n\\end{proof}\n\n\\begin{leml}\\label{pmul}\nFor every irreducible admissible smooth representations $\\Pi$ of\n$\\mathbf{J}$, one has that\n\\begin{equation*}\n \\dim {\\mathrm{Hom}}_{\\mathbf{G}}(\\Pi, \\mathbb{C}) \\,\\cdot\\, \\dim {\\mathrm{Hom}}_{\\mathbf{G}}\n (\\Pi^{\\vee},\\mathbb{C})\\leq 1.\n\\end{equation*}\n\\end{leml}\n\\begin{proof}\nThe lemma follows from Lemma \\ref{gelfand} and Lemma\n\\ref{invgeneralized} by noting that an element of the form\n\\[\n (-1,\\breve{\\mathbf g}, \\breve{\\mathbf g})\\in \\{\\pm 1\\}\\ltimes_{\\{\\pm\n1\\}}(\\breve{\\mathbf{G}}\\times_{\\{\\pm 1\\}}\\breve{\\mathbf{G}})\n\\]\nacts as an anti-automorphism on $\\mathbf{J}$.\n\\end{proof}\n\n\nLet $\\omega_\\psi$, $\\pi$ and $\\pi'$ be as in Theorem A. As in the\nproof of \\cite[Lemma 5.3]{Sun08},\n$\\omega_{\\psi}\\otimes\\pi^{\\vee}\\otimes \\pi'^{\\vee}$ is an irreducible\nadmissible smooth representation of $\\mathbf{J}$. Therefore Lemma \\ref{pmul}\nimplies that\n\\begin{equation}\\label{pin}\n \\dim {\\mathrm{Hom}}_{\\mathbf{G}}(\\omega_{\\psi}\\otimes\\pi^{\\vee}\\otimes\n\\pi'^{\\vee}, \\mathbb{C}) \\,\\cdot\\, \\dim {\\mathrm{Hom}}_{\\mathbf{G}}\n (\\omega_{\\psi}^{\\vee}\\otimes\\pi\\otimes\n \\pi',\\mathbb{C})\\leq 1.\n\\end{equation}\nBy \\cite[Theorem 1.4]{Sun09}, the two factors in the left hand side\nof (\\ref{pin}) are equal. Therefore\n\\[\n \\dim {\\mathrm{Hom}}_{\\mathbf{G}}(\\omega_{\\psi}\\otimes\\pi^{\\vee}\\otimes \\pi'^{\\vee}, \\mathbb{C})\\leq\n 1,\n\\]\nand consequently,\n\\[\n \\dim {\\mathrm{Hom}}_{\\mathbf{G}}(\\omega_{\\psi},\\pi\\otimes \\pi')\\leq\n 1.\n\\]\nThis finishes the proof of Theorem A.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe calculation of the dielectric tensor of a beam plasma system\nis a recurrent problem in plasma physics. Many efforts have been\ndedicated recently to such issue because of the Fast Ignition\nScenario for inertial thermonuclear fusion \\cite{Tabak,Tabak2005}.\nAccording to this view, the Deuterium Tritium target is first\ncompressed by means of some driver. Then, the compressed fuel is\nignited by a relativistic electron beam generated by a petawatt\nlaser shot. Such scenario implies therefore the interaction of a\nrelativistic electron beam with a plasma. This kind of\ninteraction, and its magnetized counterpart, is also relevant to\nastrophysics, in particular when investigating the relativistic\njets of microquasars \\cite{fender}, active galactic nuclei\n\\cite{zensus}, gamma ray burst production scenarios\n\\cite{Piran2004} or pulsar winds \\cite{gallant}. Theoretical works\non these subjects are usually focused on the instabilities of the\nsystem. Although many of them demands a kinetic treatment to be\nfully described, the fluid equations can set some very relevant\nguidelines, especially when the system is not too hot.\nFurthermore, it has been known for long that in the relativistic\nregime, instabilities with arbitrarily orientated wave vectors may\nbe essential\n\\cite{fainberg,Godfrey1975,Califano1,Califano2,Califano3}. One can\ntherefore figure out how some refined kinetic theory may lead to\nalmost unsolvable calculations whereas the fluid formalism is\nstill tractable. For example, a detailed description of the\ncollisional filamentation instability ($\\mathbf{k}\\perp$ beam)\nincluding the movement of the background ions plasma, and\naccounting for temperatures, was first performed through the fluid\nequations \\cite{Honda}. The very same equations were used to\nexplore the growth rate of unstable modes with arbitrarily\noriented wave vectors (with respect to the beam) when a\nrelativistic electron beam enters a plasma\n\\cite{fainberg,Califano1,Califano2,Califano3}. The results were\nfound crucial as it was demonstrated that the fastest growing\nmodes were indeed found for obliquely propagating waves, and the\nkinetic counterpart of these models has only been considered very\nrecently \\cite{BretPRE2004,BretPRE2005,BretPRL2005}. As far as the\nmagnetized case is concerned, the kinetic formalism has been\nthoroughly investigated for wave vectors parallel and normal to\nthe beam \\cite{Cary1981,Tautz2005,Tautz2006}. But\n the unstable oblique modes, which once again turn to be the most\n unstable in many cases, could only be explored through the fluid\n formalism \\cite{Godfrey1975}.\n\n It has been demonstrated that the fluid\n equations yield the same first order temperature corrections\n than the kinetic theory for oblique modes, and the roles of both\n beam and plasma parallel and perpendicular temperatures are\n retrieved \\cite{BretPoPFluide}. The fluid approximation is thus definitely a tool of\n paramount importance to deal with beam plasma instabilities.\n Additionally, it generally yields a polynomial dispersion\n equation for which numerical resolution is immediate.\n Nevertheless, even the fluid tensor can be analytically involved\n when considering arbitrarily oriented wave vectors, a guiding\n magnetic field, temperatures, and so on \\cite{BretPoPMagnet}. Indeed, on can think about\n any model based on whether the system is relativistic or\n not, collisional or not, magnetized or not, hot or cold\\ldots\n Most of these models have not been implemented yet, and each one\n should leave a quite complicated dielectric tensor.\n\n This is why\n a \\emph{Mathematica} notebook has been developed which allows for the\n symbolic calculation of the fluid tensor, once the parameters of\n the system have been set. The basic system we study here is a\n cold relativistic electron beam entering a cold magnetized plasma with return current.\n As the reader shall check, the notebook is very easy to adapt the\n different scenarios (ion beam, temperatures, pair plasma...). The paper is\n structured as follow: we start introducing the theory leading to\n the fluid dielectric tensor in section \\ref{sec:theory}. The\n \\emph{Mathematica} notebook is then explained step by step in section\n \\ref{sec:notebook}, and we show how it can be modified to include temperatures or collisions before the comments and conclusion section.\n\n\n\\section{\\label{sec:theory}Theory}\nWe consider a beam of density $n_b$, velocity $\\mathbf{V}_b$ and\nrelativistic factor $\\gamma_b=1\/(1-V_b^2\/c^2)$ entering a plasma\nof density $n_p$. Ions from the plasma are considered as a fixed\nneutralizing background, and an electron plasma return current\nflows at velocity $\\mathbf{V}_p$ such as\n$n_p\\mathbf{V}_p=n_b\\mathbf{V}_b$. The system is thus charge and\ncurrent neutralized. We do not make any assumptions on the ratio\n$n_b\/n_p$ so that the return current can turn relativistic for\nbeam densities approaching, or even equalling, the plasma one. We\nset the $z$ axis along the beam velocity and align the static\nmagnetic field along this very axis. The wave vector investigated\nlies in the $(x,z)$ plan without loss of generality\n\\cite{Godfrey1975}, and we define the angle $\\theta$ between\n$\\mathbf{k}$ and $\\mathbf{V}_b\\parallel \\mathbf{B}_0\\parallel z$\nthrough $k_z=k\\cos\\theta$ and $k_x=k\\sin\\theta$. The dielectric\ntensor of the system is obtained starting with the fluid equations\nfor each species $j=p$ for plasma electrons and $j=b$ for the beam\nones,\n\\begin{equation}\\label{eq:conservation}\n \\frac{\\partial n_j}{\\partial t}-\\nabla\\cdot (n_j\\mathbf{v}_j) =\n 0,\n\\end{equation}\n\\begin{equation}\\label{eq:force}\n \\frac{\\partial \\mathbf{p}_j}{\\partial t}+(\\mathbf{v}_j\\cdot\\nabla) \\mathbf{p}_j =\n q\\left(\\mathbf{E}+\\frac{\\mathbf{v}_j\\times \\mathbf{B}}{c}\\right),\n\\end{equation}\nwhere $\\mathbf{p}_j=\\gamma_j m\\mathbf{v}_j$, $m$ the electron mass\nand $q<0$ its charge. The equations are then linearized according\nto a standard procedure \\cite{Godfrey1975}, assuming small\nvariations of the variables according to $\\exp(i\\mathbf{k}\\cdot\n\\mathbf{r}-i\\omega t)$. With the subscripts 0 and 1 denoting the\nequilibrium and perturbed quantities respectively, the linearized\nconservation equation (\\ref{eq:conservation}) yields\n\\begin{equation}\\label{eq:conservationL}\n n_{j1} = n_{j0} \\frac{\\mathbf{k}\\cdot \\mathbf{v}_{j1}}{\\omega -\\mathbf{k}\\cdot\n \\mathbf{v}_{j0}},\n\\end{equation}\nand the force equation (\\ref{eq:force}) gives,\n\\begin{eqnarray}\\label{eq:forceL}\n &&i m \\gamma_j (\\mathbf{k}\\cdot\n\\mathbf{v}_{j0}-\\omega)\\left(\\mathbf{v}_{j1}+\\frac{\\gamma_j^2}{c^2}(\\mathbf{v}_{j0}\\cdot\n\\mathbf{v}_{j1})\\mathbf{v}_{j0}\\right)\\nonumber\\\\\n &=& q\\left(\\mathbf{E}_{1}+\\frac{(\\mathbf{v}_{j0}+\\mathbf{v}_{j1})\\times\n \\mathbf{B}_0+\\mathbf{v}_{j0}\\times\n \\mathbf{B}_1}{c}\\right),\n\\end{eqnarray}\nwhere $i^2=-1$. Through Maxwell-Faraday equations, the field\n$\\mathbf{B}_1$ is then replaced by\n$(c\/\\omega)\\mathbf{k}\\times\\mathbf{E}_1$ so that the perturbed\nvelocities $\\mathbf{v}_{j1}$ can be explained in terms of\n$\\mathbf{E}_1$ resolving the tensorial equations\n(\\ref{eq:forceL}). Once the velocities are obtained, the perturbed\ndensities can be expressed in terms of the electric field using\nEqs. (\\ref{eq:conservationL}). Finally, the linear expression of\nthe current is found in terms of $\\mathbf{E}_1$ through,\n\\begin{equation}\\label{eq:current}\n \\mathbf{J} = q\\sum_{j=p,b}\n n_{j0}\\mathbf{v}_{j1}+n_{j1}\\mathbf{v}_{j0},\n\\end{equation}\nand the system is closed combining Maxwell Faraday and Maxwell\nAmp\\`{e}re equations,\n\\begin{equation}\\label{eq:Maxwell}\n \\frac{c^2}{\\omega^2}\\mathbf{k}\\times(\\mathbf{k}\\times \\mathbf{E_1})+\\mathbf{E_1} + \\frac{4\n i\n \\pi}{\\omega}\\mathbf{J} = 0.\n\\end{equation}\nInserting the current expression from Eq. (\\ref{eq:current}) into\nEq. (\\ref{eq:Maxwell}) yields an equation of the kind\n$\\mathcal{T}(\\mathbf{E_1})=0$, and the dispersion equation reads\ndet$\\mathcal{T}=0$.\n\nThe Mathematica notebook we describe in the next section performs\na symbolic computation of the tensor $\\mathcal{T}$ and the\ndispersion equation det$\\mathcal{T}=0$, in terms of the usual\n\\cite{Ichimaru} reduced variables of the problem\n\\begin{equation}\\label{eq:param}\n\\mathbf{Z}=\\frac{\\mathbf{k}V_b}{\\omega_p},~~x=\n \\frac{\\omega}{\\omega_p},~~\\alpha=\\frac{n_b}{n_p},~~\\beta=\\frac{V_b}{c},~~\\Omega_B=\\frac{\\omega_b}{\\omega_p},\n\\end{equation}\nwhere $\\omega_p^2=4\\pi n_p q^2\/m$ is the electron plasma frequency\nand $\\omega_b=|q|B_0\/mc$ the electron cyclotron frequency.\n\n\\section{\\label{sec:notebook}\\emph{Mathematica} implementation}\nFor the most part, \\emph{Mathematica} is used to solve the\ntensorial equations (\\ref{eq:forceL}) for $\\mathbf{v}_{j1}$ and\nextract the tensors $\\mathcal{T}$ from Eqs.\n(\\ref{eq:current},\\ref{eq:Maxwell}). We start declaring the\nvariables corresponding to the wave vector, the electric field,\nthe beam and plasma drift velocities and the magnetic field,\n\n\\emph{In[1]:= }\\textbf{k = \\{kx, 0, kz\\}; E1 = \\{E1x, E1y, E1z\\};\nV0b = \\{0, 0, Vb\\}; V0p = \\{0, 0, Vp\\}; B0=\\{0, 0, $\\mathbf{m~ c~\n\\omega b\/q}$\\}; B1 = c Cross[k, E1]\/$\\mathbf{\\omega}$; vb1 =\n\\{vb1x, vb1y, vb1z\\}; vp1 = \\{vp1x, vp1y, vp1z\\};}\n\nNote that Maxwell Faraday's equation is already implemented in the\ndefinition of $\\mathbf{B1}$. The wave vector has no component\nalong the $y$ axis and the beam and plasma drift velocities only\nhave one along the $z$ axis. The guiding magnetic field is set\nalong $z$ and defined in terms of the cyclotron frequency\n$\\omega_b$. This will be useful later when introducing the\ndimensionless parameters (\\ref{eq:param}).\n\nWe then have \\emph{Mathematica} solve Eqs. (\\ref{eq:forceL}) for\nthe beam and the plasma. The left hand side of the equation is not\nas simple as in the non-relativistic case because the $\\gamma$\nfactors of the beam and the plasma modify the linearization\nprocedure. We write this part of the equations in a tensorial form\nin \\emph{Mathematica} defining the tensors \\textbf{Mp} and\n\\textbf{Mb} such as ``left hand\nside''=\\textbf{Mj}$^{-1}$.\\textbf{vj1} with,\n\n\n\\emph{In[2]:= }\\textbf{Mb=\\{\\{$\\frac{\\mathbf{i}}{\\gamma\nb(\\omega-kz Vb)}$,0,0\\},\\{0,$\\frac{\\mathbf{i}}{\\gamma b(\\omega-kz\nVb)}$,0\\},\\{0,0,$\\frac{\\mathbf{i}}{\\gamma b^3(\\omega-kz\nVb)}$\\}\\}};\n\n\\emph{In[3]:= }\\textbf{Mp=\\{\\{$\\frac{\\mathbf{i}}{\\gamma\np(\\omega-kz Vp)}$,0,0\\},\\{0,$\\frac{\\mathbf{i}}{\\gamma p(\\omega-kz\nVp)}$,0\\},\\{0,0,$\\frac{\\mathbf{i}}{\\gamma p^3(\\omega-kz\nVp)}$\\}\\}};\n\nwhere $\\mathbf{i}^2=-1$. The reader will notice that relativistic\neffects are more pronounced in the beam direction due to the\n$\\gamma^3$ factors in the $zz$ component. We now have\n\\emph{Mathematica} solve the tensorial Eqs. (\\ref{eq:forceL}). For\nbetter clarity, we first define them\n\n\\emph{In[4]:=}\\textbf{EqVb=vb1-Dot[Mb,$\\frac{\\mathbf{q}}{\\mathbf{m}}\\left(\\mathbf{E1}+\\frac{\\mathrm{Cross[\\mathbf{V0b+vb1,B0}]}}{\\mathbf{c}}\\right)$]-Dot[Mb,$\\frac{\\mathbf{q}}{\\mathbf{m}}\\left(\\frac{\\mathrm{Cross[\\mathbf{V0b,B1}]}}{\\mathrm{\\mathbf{c}}}\\right)$];}\n\n\n\\emph{In[5]:=}\\textbf{EqVp=vp1-Dot[Mp,$\\frac{\\mathbf{q}}{\\mathbf{m}}\\left(\\mathbf{E1}+\\frac{\\mathrm{Cross[\\mathbf{V0p+vp1,B0}]}}{\\mathbf{c}}\\right)$]-Dot[Mp,$\\frac{\\mathbf{q}}{\\mathbf{m}}\\left(\\frac{\\mathrm{Cross[\\mathbf{V0p,B1}]}}{\\mathrm{\\mathbf{c}}}\\right)$];}\n\nbefore we solve them,\n\n\\emph{In[6]:=}\\textbf{Vb1=FullSimplify[vb1\/.\nSolve[EqVb==0,vb1][[1]]];}\n\n\\emph{In[7]:=}\\textbf{Vp1=FullSimplify[vp1\/.\nSolve[EqVp==0,vp1][[1]]];}\n\nNote that the \\textbf{Vb}'s, with capital ``V'', store the\nsolutions of the equations whereas the \\textbf{vb}'s are the\nvariables. This is why the \\textbf{Vb}'s do not need to be defined\nat the beginning (see \\emph{In[1]}) of the notebook; they are\nimplicitly defined here.\n\nNow that we have the values of the perturbed velocities, we can\nderive the perturbed densities from Eqs. (\\ref{eq:conservationL}),\n\n\\emph{In[8]:=}\\textbf{Nb1=FullSimplify[$\\mathbf{\\omega\npb}^2\\frac{\\mathbf{m}}{4\\pi\n\\mathbf{q}^2}\\frac{\\mathbf{Dot[k,Vb1]}}{\\mathbf{\\omega-Dot[k,V0b]}}$];}\n\n\\emph{In[9]:=}\\textbf{Np1=FullSimplify[$\\mathbf{\\omega\npp}^2\\frac{\\mathbf{m}}{4\\pi\n\\mathbf{q}^2}\\frac{\\mathbf{Dot[k,Vp1]}}{\\mathbf{\\omega-Dot[k,V0p]}}$];}\n\n\n\nHere again, we prepare the introduction of the reduced variables\n(\\ref{eq:param}) by expressing the equilibrium beam and plasma\nelectronic densities in terms of the beam and plasma electronic\nfrequencies.\n\nWe can now have \\emph{Mathematica} calculate the current according\nto Eq. (\\ref{eq:current}),\n\n\\emph{In[10]:=}\n\n\\textbf{J=FullSimplify[q$\\left(\\mathbf{\\omega\npp}^2\\frac{\\mathbf{m}}{4\\pi\n\\mathbf{q}^2}\\mathbf{Vp1}+\\mathbf{\\omega\npb}^2\\frac{\\mathbf{m}}{4\\pi \\mathbf{q}^2}\\mathbf{Vb1} \\mathbf{+\nNp1 V0p+Nb1 V0b}\\right)$];}\n\nWe now have the symbolic expression of the current \\textbf{J}. In\norder to find the tensor $\\mathcal{T}$ yielding the dispersion\nequation, we need to explain first the current tensor. This is\nperformed through,\n\n\\emph{In[11]:=}\\textbf{M=}\n\\begin{displaymath}\n\\left(\n\\begin{array}{lll}\n \\mathbf{Coefficient[J[[1]],E1x]} & \\mathbf{Coefficient[J[[1]],E1y]} & \\mathbf{Coefficient[J[[1]],E1z]} \\\\\n \\mathbf{Coefficient[J[[2]],E1x]} & \\mathbf{Coefficient[J[[2]],E1y]} & \\mathbf{Coefficient[J[[2]],E1z]} \\\\\n \\mathbf{Coefficient[J[[3]],E1x]} & \\mathbf{Coefficient[J[[3]],E1y]} & \\mathbf{Coefficient[J[[3]],E1z]} \\\\\n\\end{array}\n\\right)\\mathbf{;}\n\\end{displaymath}\n\nwhich just extract the tensor elements from the expression of\n\\textbf{J}. We now turn to Eq. (\\ref{eq:Maxwell}) where we explain\nthe tensor elements of the quantity\n$c^2\\mathbf{k}\\times(\\mathbf{k}\\times\n\\mathbf{E_1})+\\omega^2\\mathbf{E_1}$,\n\n\n\\emph{In[12]:=}\\textbf{M0=$\\mathbf{c}^2$\nCross[k,Cross[k,E1]]+$\\omega^2$E1 ;}\n\n\n\\emph{In[13]:=}\\textbf{M1=}\n\\begin{displaymath}\n\\left(\n\\begin{array}{lll}\n \\mathbf{Coefficient[M0[[1]],E1x]} & \\mathbf{Coefficient[M0[[1]],E1y]} & \\mathbf{Coefficient[M0[[1]],E1z]} \\\\\n \\mathbf{Coefficient[M0[[2]],E1x]} & \\mathbf{Coefficient[M0[[2]],E1y]} & \\mathbf{Coefficient[M0[[2]],E1z]} \\\\\n \\mathbf{Coefficient[M0[[3]],E1x]} & \\mathbf{Coefficient[M0[[3]],E1y]} & \\mathbf{Coefficient[M0[[3]],E1z]} \\\\\n\\end{array}\n\\right)\\textbf{;}\n\\end{displaymath}\n\nWe can finally express the tensor $\\mathcal{T}$ defined by\n$\\mathcal{T}(\\mathbf{E})$=0 as\n\n\\emph{In[14]:=}\\textbf{T=M1+4 i $\\pi ~\\omega$ M;}\n\nAt this stage of the notebook, we could take the determinant of\nthe tensor to obtain the dispersion equation. Let us first\nintroduce the dimensionless variables (\\ref{eq:param}) through,\n\n\\emph{In[15]:=}\\textbf{T=T \/. \\{Vp $\\rightarrow -\\alpha$ Vb, kz\n$\\rightarrow \\omega \\mathbf{pp}$ Zz\/Vb, kx $\\rightarrow \\omega\n\\mathbf{pp}$ Zx\/Vb, $\\omega \\mathbf{pb}^2\\rightarrow \\alpha\n~\\omega \\mathbf{pp}^2$, $\\omega \\rightarrow \\mathbf{x} ~\\omega\n\\mathbf{pp}$, $\\omega \\mathbf{b} \\rightarrow \\Omega \\mathbf{b}\n~\\omega\\mathbf{pp}$\\}};\n\nand,\n\n\\emph{In[16]:=}\\textbf{T=T \/. \\{Vb $\\rightarrow \\beta$ c\\}}\n\n\\emph{Mathematica} leaves here some $\\omega \\mathbf{pp}$'s which\nshould simplify between each others. It is enough to perform\n\n\\emph{In[17]:=}\\textbf{T=T \/. \\{$\\omega\\mathbf{pp} \\rightarrow\n1$\\};}\n\n\nand a simple\n\n\\emph{In[18]:=}\\textbf{MatrixForm[FullSimplify[T]]}\n\ndisplays the result. The dispersion equation of the system is\neventually obtained through\n\n\\emph{In[19]:=}\\textbf{DisperEq=Det[T]}\n\nThe notebook evaluation takes 1 minute on a Laptop running a 1.5\nGHz Pentium Centrino under Windows XP Pro. This delay can be\nshortened down to 10 seconds by suppressing all the\n\\textbf{FullSimplify} routines while leaving a\n\\textbf{Simplify[T]} in entry \\emph{18}, but the final result is\nmuch less concise and readable.\n\n\\section{Comments and Conclusion}\nIn this paper, we have described a \\emph{Mathematica} notebook\nperforming the symbolic evaluation of the dielectric tensor of a\nbeam plasma system. Starting from the linearized fluid equations,\nthe notebook expresses the dielectric tensor, and eventually the\ndispersion equation, is terms of some usual dimensionless\nparameters. This notebook has been so far applied to the treatment\nof the temperature dependant non magnetized and magnetized\nproblems (see Refs \\cite{BretPoPFluide,BretPoPMagnet}). Indeed,\nthe procedure is very easy to adapt to different settings.\n\nWhen including beam or plasma temperatures, one adds a pressure\nterm $-\\nabla P_j\/n_j$ on the right hand side of the force\nequations (\\ref{eq:force}). Setting then $\\nabla P_i = 3 k_BT_i\n\\nabla n_i$ \\cite{Honda,Kruer} if dealing only with electron\nmotion, one only needs to add to the notebook entries \\emph{4} and\n\\emph{5} the terms (\\textbf{i}$^2$=-1)\n\n\\textbf{-3i Tj k $\\frac{\\mathbf{Dot[k,\nvj1]}}{\\omega-\\mathbf{Dot[k, V0j]}}$},\n\nwhere \\textbf{j=p} for the plasma, and \\textbf{b} for the beam.\nWhen considering anisotropic temperatures \\cite{BretPoPFluide},\none just needs to define a temperature tensor \\textbf{Tj} for each\nspecies \\textbf{j}, and replace the scalar product \\textbf{Tj k}\nby the tensorial one \\textbf{Dot[Tj,k]} in both entries. Of\ncourse, a correct treatment of electromagnetic instabilities\ngenerally requires a kinetic formalism instead of a fluid one.\nHowever, kinetic calculations cannot be systematically entrusted\nto \\emph{Mathematica}, as is the case here. The reason why is that\nthe relativistic factors $\\gamma$ encountered in the kinetic\nquadratures are coupling the integrations along the three\ncomponents of the momentum. According to the distribution\nfunctions considered, the quadratures may be calculable through\nsome ad hoc change of variables, if they can be calculated at all.\nAt any rate, the process cannot be systematized enough for\n\\emph{Mathematica} to handle it.\n\nAs far as the magnetic field is concerned, its direction can be\nchanged from entry \\emph{1} without any modification of the next\nentries. When dealing with the motion of ions, or even with one of\nthese pair plasmas involved in the pulsar problems\n\\cite{GedalinPRL}, one just need to modify the conservation and\nforce equations according to the properties of the species\ninvestigated. It is even possible to add more equations to account\nfor more species because the resolution involves only the force\nand the conservation equations of one single specie at a time\nbefore the perturbed quantities merge together in entry \\emph{10}\nto compute the current \\textbf{J}.\n\nThe notebook can thus be easily adapted to different settings and\nallows for a quick symbolic calculation of the dielectric tensor\nand the dispersion equation, even for an elaborated fluid model.\n\n\n\\section{Acknowledgements}\nThis work has been achieved under projects FTN 2003-00721 of the\nSpanish Ministerio de Educaci\\'{o}n y Ciencia and PAI-05-045 of\nthe Consejer\\'{i}a de Educaci\\'{o}n y Ciencia de la Junta de\nComunidades de Castilla-La Mancha.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\nScientific portals such as PubMed, Google Scholar, Microsoft Academic Search, \nCiteSeer$^{\\tt x}$, and ArnetMiner provide access to scholarly publications and \ncomprise indispensable resources for researchers who search for literature \non specific subject topics. In addition, data mining applications such as \ncitation recommendation~\\cite{wsdm11he}, expert search~\\cite{ijcai07balog},\ntopic trend detection~\\cite{kdd06wang,cikm09he}, and author influence modeling~\\cite{ijcai11kataria}\ninvolve web-scale analysis of up-to-date research collections. While\nacademics and researchers\\footnote{\\scriptsize In this paper, we use \nthe terms ``researchers\/authors\/scholars\" and ``research documents\/papers\/publications\" interchangeably. \nWe also use (academic) homepages to refer to professional homepages maintained by scholars and {``Scholarly\/Academic Web\" to \nrefer to sections of the Web (for example, university websites and research centers) that cater to scholarly pursuits.}} \n continue to produce large numbers of scholarly documents worldwide, acquisition of \nresearch document collections becomes a challenging task for digital libraries.\n\nIn contrast with commercial portals \n(such as the ACM digital library)\nthat rely on clean and structured publishing sources for their collections, open-access, autonomous systems such as CiteSeer$^{\\tt x}$ and ArnetMiner \nacquire and index freely-available research articles on the Web~\\cite{infoscale06li,kdd08tang}. \nResearchers' homepages and paper repository URLs are crawled and processed \nperiodically for maintaining the research collections in these portals. Needless to say, \nthese repositories are incomplete since the crawl seed lists cannot \nbe comprehensive in face of the ever changing Scholarly Web. Not only do new authors and publication venues\nemerge, but also existing researchers may stop publishing or change universities resulting in\noutdated seed URLs. \\textit{Given this challenge, how can we automatically augment crawl seed lists\nfor a scientific digital library?} \n\n\nWeb search has been a constant topic of investigation for\nIR, ML, and AI research groups since several years. Current \nWeb search engines feature state-of-the-art\ntechnologies, ranking algorithms, syntax, personalization and localization features along\nwith efficient infrastructure and programmable APIs making them invaluable tools\nto access and process the otherwise intractable Web. Despite these attractive advancements, to the best of \nour knowledge, search-driven methods are yet to be investigated as alternatives\nto crawl-based approaches for acquiring documents in digital libraries. In this paper, \nwe address \nthis gap in the context of open-access, scientific digital libraries. We propose\na novel Search\/Crawl framework, describe its components and present experiments \nshowcasing its potential in acquiring research documents.\n\nTo motivate our framework, we recall how a\nWeb user typically searches for research papers or authors~\\cite{nips02richardson,cikm08serdyukov}. \nAs with regular document search, a user typically issues Web search queries \ncomprising of representative keywords or paper titles\nfor finding publications on a topic. Similarly, if the author is known, \na ``navigational query\"~\\cite{sigir02broder} may be employed to locate the homepage\nwhere the paper is likely to be hosted. Indeed, according to\nprevious studies, researchers provide access to their papers (when possible) to improve their visibility and \ncitation counts making researcher homepages {a likely hub for locating\nresearch papers}~\\cite{nature01lawrence,tweb15gollapalli}.\n\\begin{figure*}[!htp]\n\\centering\n\\hspace*{-0.65cm}\n\\includegraphics[scale=0.35]{anecdote_ijcai.eps}\n\\caption{\\small An anecdotal example for illustration (searches performed on Jan 26, 2016).}\n\\label{fig:anecdotalsearch}\n\\end{figure*}\n\nGiven previous knowledge \nin academic browsing, scholars are often able to accurately locate the correct research \npapers or academic homepages from the Web search results using hints\nfrom the titles, search summaries (or snippets) and the URL strings. To illustrate this process, Figure~\\ref{fig:anecdotalsearch} shows an anecdotal example of a \nsearch using Google for the title and authors of a paper published at IJCAI last year, ``Maximum Satisfiability using Cores and Correction Sets'' by Nikolaj Bjorner and Nina Narodytska. For the top-$5$ results shown for the paper title query (set 1), \nfour of the five results are research papers on the topic. The document\nat the Springer link is not available for free whereas the last document corresponds to\ncourse slides.\nFor the homepage URLs identified from author name search results (from sets 2 and 3), namely: \\\\\n\\begin{scriptsize}\n\\texttt{\\textbf{http:\/\/www.cse.unsw.edu.au\/~ninan\/}} \\\\\n\\texttt{\\textbf{http:\/\/research.microsoft.com\/en-us\/people\/nbjorner\/}} \\\\\n\\texttt{\\textbf{http:\/\/theory.stanford.edu\/people\/nikolaj\/}} \n\\end{scriptsize}\n\\\\\nwe found $55$ documents, $46$ of which correspond to research publications.\nThis anecdotal search example highlights the immense potential of Web search \nfor retrieving research papers and seed URLs that can be crawled for research papers.\n\nOur Search\/Crawl framework mimics precisely the above search and scrutinize approach\nadopted by Scholarly Web users. Freely-available information from the Web for specific subject disciplines\\footnote{\n\\scriptsize For example, from bibliographic listsings such as DBLP.}\nis used to frame title and author name queries in our framework. The two control flow paths\nfor obtaining research papers are highlighted in \nFigure~\\ref{fig:schematic}. Research paper titles are used as queries \nin \\textbf{Path 1}. The documents resulting from this search are classified \nwith a paper classifier based on Random Forests~\\cite{mlj01breiman}. Author names comprise the queries \nfor Web search in \\textbf{Path 2}, the\nresults of which are filtered using a \nhomepage identification module trained using the RankSVM algorithm~\\cite{kdd02joachims}.\nThe predicted academic homepages serve as seeds for \nthe crawler module that obtains all documents upto a depth $2$ starting from the seed URL. \nThe paper identification module is once again employed\nto retain only those documents relevant to a scientific digital library among the crawled documents.\nWe summarize our contributions below:\n\\begin{itemize}\n\\item We propose a novel framework based on search-driven methods to \nautomatically acquire research documents \nfor scientific collections. To the best of our knowledge, we are the first to\nuse ``Web Search\" to obtain seed URLs for initiating crawls \nin an open-access digital library.\n\\item Our Search\/Crawl framework interleaves several existing and new modules.\nWe extend existing research on academic document classification to identify research \npapers among documents. Next, we design a novel homepage identification module,\na crucial component\n for our framework, that\nuses several features based on webpage titles, URL strings, and terms in the \nresult snippet to identify researcher homepages from the results of {author name search}. The identified \nhomepages form seeds for our Web crawler.\n\\item We provide a thorough evaluation of both the paper and homepage identification components \nusing various publicly-available datasets. Our proposed features \nattain state-of-the-art performance on both these tasks.\n\\item Finally, we perform a large-scale, first-of-its-kind experiment using $43,496$ research paper \ntitles and $32,816$ author names from Computer Science. We not only recovered approximately $75$\\% of the papers\ncorresponding to the research paper title queries but were also able to collect about $0.665$ million \nresearch documents overall with our framework. These impressive yields showcase our \nWeb-search driven methods to be highly effective for obtaining and maintaining up-to-date\ndocument collections in open-access digital library portals.\n\\end{itemize}\n\\begin{figure}[!htbp]\n\\centering\n\\includegraphics[height=2.5in, width=3in]{schematic.eps}\n\\caption{\\small Schematic Diagram of our Search\/Crawl framework.}\n\\label{fig:schematic}\n\\end{figure}\n\nWe provide details of our paper and homepage identification modules in Section~\\ref{sec:methods}. In Section~\\ref{sec:expts}, we describe our experimental setup, results,\nand findings. We briefly summarize closely-related work in Section~\\ref{sec:related} and present\nconcluding remarks in Section~\\ref{sec:conclude}.\n\n\n\\section{AI Components in Our Framework}\n\\label{sec:methods}\nThe accuracy and efficiency our \nSearch\/Crawl framework is contingent\non the accuracies of two components: (1) the homepage identifier and (2)\nthe paper classifier. \n\n\\textbf{Homepage Identification}:\nAcademic\nhomepages, known to link to research papers~\\cite{nature01lawrence}, form potential seed URLs for initiating\ncrawls in digital libraries. For our Search\/Crawl framework to be effective and efficient, it is imperative to identify \nthese pages from the search results of author name queries. \nIdentifying researcher homepages among other types of webpages can be treated as an instance of the \nwebpage classification problem with the underlying classes: homepage\/non-homepage~\\cite{tweb15gollapalli}. \nHowever, given the Web search setting, the non-homepages retrieved in response to an author name query can be expected to be diverse with\nwebpages ranging from commercial websites such as LinkedIn, social media websites such as Twitter and Facebook,\npublication listings such as Google Scholar, Research Gate, and several more. To handle this aspect, we draw ideas from the recent developments in \nWeb search ranking and frame homepage identification as a ranking problem.\n\nGiven a set of webpages in response to a query, our objective\nis to rank homepages better, i.e., top ranks, relative to other types of webpages, capturing\nour preference among the webpages. For example, consider\na name query ``John Blitzer\" and let \nthe results in response to web search be:\\\\\n\\begin{small}\n\\begin{tabular}{ll}\n\\hline\nRank & URL \\\\\n\\hline\n1 & research.google.com\/pubs\/author14735.html \\\\\n2 & john.blitzer.com \\\\\n3 & https:\/\/www.linkedin.com\/pub\/john-blitzer\/5\/606\/425 \\\\\n4 & http:\/\/dblp.uni-trier.de\/pers\/hd\/b\/Blitzer:John \\\\\n\\hline\n\\end{tabular}\n\\end{small}\n\\\\\n\nSuppose ``john.blitzer.com\" is known to be the correct homepage and we are not interested\nin other webpages. This desirable property can be expressed via\nthree preference pairs among the ranks: $p_2>p_1, p_2>p_3, p_2>p_4$\nwhere $p_i$ refers to the webpage at rank $i$. Note that, we do not express preferences\namong the non-homepages $p_1$, $p_3$, and $p_4$. Preference information such as the\nabove is modeled through appropriate objective functions in \nlearning to rank approaches~\\cite{ftir09liu}. For example, a RankSVM minimizes the\nKendall's $\\tau$ measure based on the preferential ordering\ninformation present in training examples~\\cite{kdd02joachims}.\n\nLearning to rank methods were heavily investigated for capturing user\npreferences in clickthrough logs of search engines as well as in NLP tasks such as summarization and keyphrase extraction~\\cite{ltrbook11li}.\nNote that, unlike classification approaches that independently model both positive (homepage) and negative (non-homepage) \nclasses, \nwe are modeling instances in relation with each other \nwith preferential ordering~\\cite{kdd02joachims,icml05burges,ijcai15wan}.\nWe show that the ranking approach out-performs classification approaches for homepage identification\nin Section~\\ref{sec:expts}. We use the following feature types:\n\\begin{enumerate}\n\\item \\textbf{URL Features}: Intuitively, the URL strings of academic \nhomepages can be expected to contain or not contain certain\ntokens. For example, a homepage URL is less likely to be hosted on domains such as ``linkedin\" and ``facebook\".\nOn the other hand, terms such as\n``people\" or ``home\" can be expected to occur in the URL strings of homepages (example homepage URLs in Figure~\\ref{fig:anecdotalsearch}) .\nWe tokenize the URL strings based on the ``slash (\/)\" separator\nand the domain-name part of the URL based on the ``dot ($.$)\" separator to extract \nour URL and DOMAIN feature dictionaries. \n\\item \\textbf{Term Features}: Current-day search engines present Web search\nresults as a ranked list where each webpage is indicated by its HTML title, URL string as well\nas a brief summary of the content of the webpage (also known as the ``snippet\"). \nPrevious research has shown that\nusers are able to make appropriate ``click\" decisions during \nWeb searches based on this presented information~\\cite{nips02richardson,sigir04granka}. \nWe posit that users of Scholarly Web are able to identify homepages among the search results based on the term hints in titles\nand snippets (for example, ``professor\", ``scientist\", ``student\") and capture these keywords in TITLE and SNIPPET dictionaries.\n\\item \\textbf{Name-match Features}: These features capture the common observation that \nresearchers tend to use parts of their names in the URL strings of their homepages~\\cite{icdm07tang,tweb15gollapalli}.\nWe specify two types of match features: (1) a boolean feature that indicates whether any part of the author name matches a token \nin the URL string, and\n(2) a numeric feature that indicates the extent to which name tokens overlap with the (non-domain part of) URL string given by the fraction: $\\frac{\\#{\\tt matches}}{\\#{\\tt name tokens}}$.\nFor the example author name ``Soumen Chakrabarti\" and the URL string: \\texttt{\\small \\textbf{www.cse.iitb.ac.in\/$\\sim$soumen}}, \nthe two features have values ``true\" and $0.5$, respectively.\n\\end{enumerate}\n\nThe dictionary sizes for the above feature types based on our training datasets (Section~\\ref{sec:expts})\nare listed below: \\\\\n\\begin{center}\n\\begin{tabular}{ll}\n\\hline\nFeature Type & Size\\\\\n\\hline\nURL+DOMAIN term features & 2025 \\\\\nTITLE term features & 19190 \\\\\nSNIPPET term features & 25280 \\\\\nNAME match features & 2 \\\\\n\\hline\n\\end{tabular}\t\n\\end{center}\n\n\\textbf{Paper Classification}: Recently, Caragea et al. \\shortcite{iaai16caragea} studied\nclassification of academic documents \ninto six classes: Books, Slides, Theses, Papers, CVs, and Others. \nThey experimented with\nbag-of-words from the textual content of the documents (BoW), tokens in the document URL string (URL),\nand structural features of the document (Str) and showed that a small \nset of structural features are highly indicative of \nthe class of an academic document. Their set of $43$ structural features\nincludes features such as size of the file, number of pages in the document, average number of words\/lines\nper page, phrases such as ``This thesis\", ``This paper\" and the relative \nposition of the Introduction and Acknowledgments sections.\\footnote{\\scriptsize\nWe refer\nthe reader to~\\cite{iaai16caragea} for a complete listing of features used\nfor training this classifier.}\n\nWe found that these structural features continue to perform\nvery well on our datasets (Section~\\ref{sec:expts}) with precision\/recall values in the ranges of $90+$.\nTherefore, we directly employ their features for training the \npaper classification module in our framework. However, since\nwe are not interested in other types of documents and because binary\ntasks are considered easier to learn than multiclass tasks~\\cite{mlboook06bishop}, \nwe re-train the classifiers for the two-class setting: papers\/non-papers. \n\n\\section{Datasets and Experiments}\n\\label{sec:expts}\nIn this section, we describe our experiments on homepage\nidentification and paper classification along with their performance\nwithin the Search\/Crawl \npaper acquisition framework.\nOur datasets are summarized in Table~\\ref{tab:dspaperauthor} and described below:\n\\begin{enumerate}\n\\item For evaluating homepage finding using author names, \nwe use the\nresearcher homepages from DBLP, the bibliographic reference\nfor major Computer Science publications.\\footnote{\\scriptsize http:\/\/dblp.uni-trier.de\/xml\/}\nIn contrast to previous works that use this dataset to train homepage classifiers\non academic websites~\\cite{tweb15gollapalli}, in\nour Web search scenario, \nthe non-homepages from the search results of a name query\nneed not be restricted to academic websites.\nExcept the true homepage, all other webpages \ntherefore correspond to negatives. We collected the DBLP dataset as follows: \nUsing the\nauthor names as queries, we perform Web search \nand scan the \ntop-$k$\nresults in response to each query.\\footnote{\\scriptsize We used the Bing API for \nall Web search experiments and retrieve the top-$10$ results. All queries are ``quoted\" to \nimpose exact match and ordering of tokens and the filetype syntax was used to \nretrieve PDF or HTML files as applicable.}\nIf the true homepage from DBLP is listed among\nthe top results, this URL and the others in the set of Web results can be used \nas training instances. We used RankSVM\\footnote{\\scriptsize http:\/\/svmlight.joachims.org\/}\nfor learning a ranking function for author name search. In this model,\nthe preference among the search results for a query can be indicated by simply assigning the\nranks ``1\" and ``2\" respectively to the true and remaining results. \nFor classification algorithms, we directly use \nthe positive and negative labels for these webpages. We were able to locate homepages for $4255$ authors in the top-$10$ results \nfor the author homepages listed in DBLP.\n\\begin{table}[htp]\n\\centering\n\\begin{small}\n\\begin{tabular}{|llr|}\n\\hline\n\\textbf{Dataset} & & \\\\ \n\\hline\nResearch Papers &(Train) &\t960(T) 472(+) \\\\\n\t\t&(Test) & 959(T) 461(+) \\\\\n\\hline\nDBLP Homepages &\\multicolumn{2}{r|} {42,548(T) 4,255(+)} \\\\\n\\hline\nCiteSeer$^x$ & \\multicolumn{2}{l|} {43,496 (Titles), 32,816(Authors)} \\\\\n\\hline\n\\end{tabular}\n\\end{small}\n\\caption{\\small Summary of datasets used in experiments. The numbers of total and positive instances are shown \nusing (T) and (+), respectively, for the labeled datasets.}\n\\label{tab:dspaperauthor}\n\\end{table}\n\\item Caragea et al. \\shortcite{iaai16caragea} randomly sampled\ntwo independent sets of approximately $1000$ documents each from the crawl\ndata of CiteSeer$^{\\tt x}$.\nThese sets, called Train and Test, were manually labeled\ninto six classes: Paper, Book, Thesis, Slides, Resume\/CV, and Others. \nWe transform the documents' labels as the binary labels, Paper\/Non-paper, and use these datasets directly in our experiments.\n\\item For our third dataset,\nwe extracted research papers from the \npublication venues listed in Table~\\ref{tab:venues} from the\nCiteSeer$^{\\tt x}$ scholarly big dataset \\cite{ecir14caragea}, in which \npaper metadata (author names, venues, and paper titles) are mapped to entries in DBLP to ensure \na clean collection.\\footnote{\\scriptsize Machine learning-based modules are used for extracting titles, venues, and authors of a paper \nin CiteSeer$^{\\tt x}$ thus resulting in occasional erroneous metadata.} Overall, we obtained a set of $43,496$ paper titles,\nauthors ($32,816$ unique names) for evaluating \nour Search\/Crawl framework at a large scale.\n\n\n\\end{enumerate}\n\n\\begin{table}[!htp]\n\\begin{center}\n\\begin{scriptsize}\n\\begin{tabular}{|l|}\n\\hline\t\n{Total \\# of research papers: 43,496, \\#authors (unique names): 32,816}\\\\\n\\hline\nNIPS (5211), IJCAI (4721), ICRA (3883), ICML (2979), \\\\ \nACL (2970) , VLDB (2594), CVPR (2373), AAAI (2201), \\\\ \nCHI (2030), COLING (1933), KDD (1595), SIGIR (1454), \\\\\nWWW (1451), CIKM (1408), SAC (1191), LREC (1128), SDM (1111), \\\\\nEMNLP (920), ICDM (891), EACL (760), HLT-NAACL (692) \\\\\n\\hline\n\\end{tabular}\n\\end{scriptsize}\n\\end{center}\n\\caption{\\small Conference venue\/\\#papers in the CiteSeer$^{\\tt x}$ dataset.\n}\n\\label{tab:venues}\n\\end{table}\n\nWe use the standard measures Precision, Recall, and F1 for summarizing the \nresults of author homepage identification and paper classification~\\cite{irbook08manning}. \nUnlike classification where we\nconsider the true and predicted labels for each instance (webpage), in RankSVM the prediction is \nper query~\\cite{kdd02joachims}. That is, the results with respect to a query are assigned ranks based on scores\nfrom the\nRankSVM and the result at rank-1 is chosen as the predicted\nhomepage.\nThe implementations in Weka~\\cite{weka}, Mallet~\\cite{mallet} and SVMLight~\\cite{svmlight} were used for models' training and evaluation.\n\n\\subsection{Author Homepage Finding}\nWe report the five-fold cross-validation performance of the homepage identification module\ntrained using various classification modules and RankSVM\nin Table~\\ref{tab:asresults}. \nThe best performance obtained with all features described in Section~\\ref{sec:methods}\non the DBLP dataset after tuning \nthe learning parameters (such as C for SVMs), is shown in this table.\nRankSVM captures the relative preferential ordering among search results and\nperforms the best in identifying the correct author homepage in response to a query. \nA possible reason for the lower performance of the classification approaches \nsuch as Binary SVMs, Na\\\"ive Bayes, and Maximum Entropy is \nthat they model the positive and negative instances\nindependently and not in relation to one another for a given query. Moreover, the\ndiversity in webpages among the negative class is ignored and they \nare modeled uniformly as a single class in these methods.\n\n\\begin{table}[!htp]\n\\begin{center}\n\\begin{small}\n\\begin{tabular}{|l|c|c|c|}\n\\hline\n\\textbf{Method} & \\textbf{Precision} & \\textbf{Recall} & \\textbf{F1} \\\\\n\\hline\nNa\\\"ive Bayes & 0.4830 & 0.9239 & 0.63432 \\\\\nMaxEnt & 0.8207 & 0.8002 & 0.8102 \\\\\nBinary SVM & 0.8353 & 0.8149 & 0.8249 \\\\\nRankSVM & \\textbf{0.8900} & \\textbf{0.8900} & \\textbf{0.8900} \\\\\n\\hline\n\\end{tabular}\n\\end{small}\n\\end{center}\n\\caption{\\small Classifier and RankSVM performances on DBLP dataset.}\n\\label{tab:asresults}\n\\end{table}\n\nWe point out that false positives are not very critical in our Search\/Crawl framework.\nIncluding an incorrectly predicted homepage as\na seed URL may result in crawling irrelevant documents and extra processing load. However, \nthese documents are subsequently filtered out by our paper classifier.\n\n\\begin{table}[!htp]\n\\centering\n\\begin{small}\n\\begin{tabular}{l r || l r}\n\\hline\n\\textbf{FeatureType} & \\textbf{Feature} & \\textbf{FeatureType} & \\textbf{Feature} \\\\ \\hline\nNAME & fracMatch & TITLE & university \\\\ \nDOMAIN & com & SNIPPET & computer \\\\ \nNAME & hasMatch & TITLE & homepage \\\\ \nTITLE & home & SNIPPET & university \\\\ \nTITLE & page & TITLE & linkedin \\\\ \nSNIPPET & professor & SNIPPET & science \\\\ \nDOMAIN & edu & SNIPPET & discover \\\\ \nSNIPPET & view & URL & author \\\\ \nSNIPPET & department & SNIPPET & linkedin \\\\ \nSNIPPET & profile & SNIPPET & professionals \\\\ \\hline \n\\end{tabular}\n\\end{small}\n\\caption{\\small The top-$20$ features ranked based on Information Gain.}\n\\label{tab:asigfeats}\n\\end{table}\nTable~\\ref{tab:asigfeats} shows the top features based on information gain values~\\cite{jmlr03forman03}.\nThese features make intuitive sense; for instance, a researcher homepage is likely to have parts of \nthe researcher name mentioned on it along with terms like ``home\" and ``page\" in the HTML title. \nSimilarly, webpages typically ending in ``.com\" or \nhaving ``linkedin\" in their description are unlikely to be homepages.\n\t\n\\subsection{Research Paper Identification}\nThe results of paper classification are summarized in Table~\\ref{tab:perfpaper}.\nWe directly used the feature sets proposed by Caragea et al. \\shortcite{iaai16caragea}\nand tested various classifiers including Na\\\"ive Bayes, Support Vector Machines and \nRandom Forests. All models are trained on the ``Train'' dataset. The parameters of each model are tuned \nthrough cross-validation on the ``Train\" dataset and the classification performance evaluated\non the ``Test\" dataset. {The results of various\nfeatures sets using a Random Forest for the ``paper\" class in the binary setting\nare shown in Table~\\ref{tab:perfpaper}. We also show\nthe performance on the ``paper\" class with the \nmulticlass setting and the weighted averages of all measures over all classes for\nboth the settings in this table. \nThe best classification performance is obtained using a Random Forest trained on\nstructural features with the overall performance being\nsubstantially better in the two-class setting rather than the multiclass setting.}\n\\begin{table}[htp]\n\\centering\n\\begin{small}\n\n\\begin{tabular}{|l|c|c|c|}\n\\hline\n{Feature}&{Precision}&{Recall}&{F1}\\\\\n\\hline\nBoW (P)& 0.86 & 0.92 & 0.889 \\\\\nURL (P)& 0.729 & 0.729 & 0.729 \\\\\nStr\/Binary (P)& \\textbf{0.933} & \\textbf{0.967} & \\textbf{0.950} \\\\\nStr\/Multiclass (P) & 0.918 & 0.965 & 0.941 \\\\\n\\hline\nStr\/Binary (A)& \\textbf{0.952} & \\textbf{0.951} & \\textbf{0.951} \\\\\nStr\/Multiclass (A) & 0.893 & 0.902 & 0.892 \\\\\n\\hline\n\\end{tabular}\n\\end{small}\n\\caption{\\small Classification performance on the test dataset. `P\/A' indicate performances for ``Paper\"\/``All\" classes.}\n\\label{tab:perfpaper}\n\\end{table}\n\n\\subsection{Search\/Crawl Experiments}\n\\begin{table*}[!htp]\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\n\\#Queries & \\#PDFs & \\#Papers & \\#UniqueTitles & \\#Matches\\\\\n\\hline\n43,496 titles (Path 1)& 322,029 & 213,683 & 91,237 & 32,565 \\\\\n32,816 names (Path 2) & 665,661 & 452,273 & 204,014 & 17,627\\\\\n\\hline\n\\end{tabular}\n\\caption{\\small \\#Papers obtained through the two paths in our Search\/Crawl framework.}\n\\label{tab:csxresults}\n\\end{center}\n\\end{table*}\nFinally, we evaluate the two AI components in \npractice within our Search\/Crawl framework\nusing the CiteSeer$^{\\tt x}$ dataset. To this end, for \\textbf{Path 1}, we use\nthe $43,496$ paper titles as search queries. Structural features\nextracted \nfrom the resulting PDF documents of each search are used\nto identify research documents with our paper classifier. For \\textbf{Path 2}, the $32,816$ author names are used as queries. The\nRankSVM-predicted homepages from the results of each query are crawled for PDF documents up to a depth of \n$2$.\\footnote{\\scriptsize We used the wget utility for our crawls (https:\/\/www.gnu.org\/software\/wget\/)} \nOnce again, the paper classifier is employed to\nidentify research documents among the crawled documents. \n\nThe number of PDFs and papers found through the two paths in \nour proposed Search\/Crawl framework are shown in Table~\\ref{tab:csxresults}. \nSince our dataset is based on CiteSeer$^{\\tt x}$, we removed all paper search results\nthat point to CiteSeer$^{\\tt x}$ URLs for a fair evaluation. The number of papers \nthat we could obtain from the original $43,496$ collection through both the paths \nare shown in the last column of this table. We use the title\nand author names available in the dataset to look up the first page of the PDF document\nfor computing this match.\n\nWe are able to obtain $75\\%$ ($\\frac{32565}{43496}$)\nof the original titles through \\textbf{Path 1} compared to the $40\\%$ \n($\\frac{17627}{43496}$) through \\textbf{Path 2} (column $5$ in Table~\\ref{tab:csxresults}).\n{In general,\ngiven that paper titles contain representative keywords~\\cite{emnlp14caragea,Litvak2008},\nif they are available online, a Web\nsearch with appropriate filetype filters\nis a successful strategy for finding them. The high percentage\nof papers found along \\textbf{Path 2} confirms previous findings that\nresearchers tend to link to their papers via their homepages~\\cite{nature01lawrence,tweb15gollapalli}.}\n\nIntuitively, the overall yield can be expected to be higher through \\textbf{Path 2}.\nOnce an author homepage is reached, other research papers linked to this page \ncan be directly obtained. Indeed, as shown in columns $2$ and $3$ of Table~\\ref{tab:csxresults}, \nthe numbers of PDFs as well as classified papers are significantly higher along \\textbf{Path 2}. \nCrawling the predicted homepages of the $32,816$ authors we obtain approximately\n$14$ research papers per query on average ($\\frac{452273}{32816}=13.78$). In contrast,\nexamining only the top-$10$ search results along \\textbf{Path 1}, we obtain $5$ research \ndocuments per query ($\\frac{213683}{43496}=4.91$). \n\nWe used the CRF-based title extraction tool for research papers, ParsCit~\\footnote{\\scriptsize http:\/\/aye.comp.nus.edu.sg\/parsCit\/} \nto extract the titles of the research papers obtained from both the paths. \n{The number of extracted unique titles are shown in column $4$ of Table~\\ref{tab:csxresults}. \nThe overlap in the two sets of titles is $28,374$. Compared to the overall yields\nalong \\textbf{Path 1} and \\textbf{Path 2},\nthis small overlap indicates that the two paths are capable of reaching different sections of the Web and play\ncomplementary roles in our framework. For example, the top-$20$ domains of the URLs from which \nwe obtained research papers along \\textbf{Path 1} are shown in Table~\\ref{tab:topdomainsp1}. Indeed, \nvia Web search we are able to reach a wide range of domains. This is unlikely in crawl-driven methods\nwithout an exhaustive list of seeds since only\nlinks up to a specified depth from a given seed are explored~\\cite{irbook08manning}.}\n\nTo summarize, using about $0.076$ million queries ($43,496+32,816$) in our framework, we are able to build a\ncollection of $0.665$ million research documents ($213,683+452,273$) and \n$0.267$ million unique titles\n($91,237+204,014-28,374$). About $32-33\\%$ of the obtained documents\nare ``non-papers\" along both the paths. Scholarly Web is known to contain a variety of documents\nincluding project proposals, resumes, and course materials~\\cite{jis06ortega}. \nIndeed, some of these documents may include the exact paper titles and show up in paper search results as well\nas be linked to author homepages. In addition, \nusing incorrectly-predicted homepage as seeds may result in ``bad\" documents.\n\\begin{table}[!hp]\n\\centering\n\\begin{small}\n\\begin{tabular}{|l|}\n\\hline\nedu (71,139), org (47,272), net (20,552), com (19,178), de (5,424)\\\\\nuk (5,065), fr (3,770), ca (3,651), it (2,647), gov (2,130), \\\\\nnl (1,891), cn (1,777), jp (1,673), au (1,655), cc (1,489), \\\\ \nch (1,431), sg (1,282), in (1,209), il (1,206), es (1,144) \\\\\n\\hline\n\\end{tabular}\n\\end{small}\n\\caption{\\small The top-20 domains from which papers were obtained along Path-1 of our framework.}\n\\label{tab:topdomainsp1}\n\\end{table}\n\n\\textbf{Sample Evaluation.} Given the size of the CiteSeer$^{\\tt x}$ dataset and the \nlarge number of documents obtained via the Search\/Crawl framework (Table~\\ref{tab:csxresults}), it is\nextremely labor-intensive to manually examine all documents resulting from this experiment. However,\nsince our classifiers and rankers are not $100\\%$ accurate and we only\nexamine the top-$k$ results from the search engine, we need an estimate of how many papers\nwe are able to obtain via our Search\/Crawl approach among those\nthat are actually obtainable on the Web. We randomly selected $10$ titles from the CiteSeer$^{\\tt x}$ dataset and their \nassociated set of $78$ authors and inspected \nall PDFs that can be obtained via our search\/crawl framework manually.\nThat is, we searched for the selected paper titles and manually examineed and annotated the resulting PDFs.\nSimilarly, the correct homepages of the $78$ authors were obtained by searching the Web\nand manually examining the resulting webpages. The correct homepages were crawled (to depth 2)\nfor PDFs and the resulting documents were manually labeled.\n\nWe were able to locate $49$ correct homepages of the $78$ authors in this manual experiment. Crawling these homepages resulted\nin $2116$ PDFs out of which $1418$ were found to be research papers.\nOur Search\/Crawl framework that crawls predicted homepages for the $78$ authors \nand uses paper classifier\npredictions to identify research papers was able to \nobtain $1291$ research papers. Out of these documents, $1104$ match with\nthe intended set of $1418$ papers. Thus, we are able to \nobtain approximately $\\textbf{78}\\%$ of the intended set of papers along with an additional $187$ new ones.\nPaper search using titles results in $59$ PDFs out of which $33$ are true papers.\nOur paper classifier obtains a precision\/recall of $\\textbf{84}$\\%\/$\\textbf{97}$\\%, predicting $32$ out of these $33$ papers correctly\nand $38$ papers overall. \n\n\\section{Related Work}\n\\label{sec:related}\nHomepage finding and document classification are \nvery well-studied problems. \nDue to space constraints, we refer the reader to \nthe TREC 2001 proceedings\\footnote{\\scriptsize http:\/\/trec.nist.gov\/proceedings\/proceedings.html}\nand the comprehensive\nreviews of the feature representations, methods, and results\nfor various text\/webpage classification problems~\\cite{cs02sebastiani,cs09qi}.\n\nThough homepage finding in TREC did not\nspecifically address researcher homepages, this track \nresulted in various state-of-the-art machine learning systems\nfor finding homepages~\\cite{spire02xi,tis03upstill,icadl06wang}. \nAmong works focusing specifically on researcher homepages, both Tang et al.~\\shortcite{icdm07tang} and Gollapalli et al.~\\shortcite{tweb15gollapalli}\ntreat homepage finding as a binary classification task and use\nvarious URL and content features. Ranking methods were \nexplored for homepage finding using the top terms\nobtained from topic models~\\cite{sigireos11gollapalli}.\n\nIn the context of scientific digital libraries,\ndocument classification into\nclasses related to subject-topics (for example, ``machine learning\", ``databases\")\nwas studied previously~\\cite{icml03lu,emnlp15caragea}.\nOften bag-of-words features as well as topics extracted using LDA\/pLSA are used to represent\nthe underlying documents in these works. Structural features, on the other\nhand, are popular in classifying and clustering semi-structured XML documents~\\cite{icpr08ghosh,aireview13asghari}. \n\nIn contrast with existing work,\nwe investigate features from web search engine results\nand formulate researcher homepage identification as a learning to rank task. In addition,\nwe are the first to interleave various AI components with \nexisting Web search and crawl modules to build an efficient paper acquisition framework.\n\n\\section{Conclusions}\n\\label{sec:conclude}\nWe proposed a search-driven framework for automatically acquiring research\ndocuments on the Web as an alternative to\ncrawl-driven methods adopted in current open-access digital libraries.\nOur framework crucially depends on accurate paper classification and researcher \nhomepage identification modules. To this end, we discussed features for \nthese modules and showed experiments illustrating their\nstate-of-the-art performance. In one experiment using a large collection\nof about $0.076$ million queries, our framework was able to automatically\nacquire a collection of approximately $0.665$ million research documents. These\nresults showcase the potential of our proposed framework in \nimproving scientific digital library collections.\n{For future work, apart from improving the accuracies of individual components in our framework, we will focus on \nincluding other document formats (for example, .ps and zipped files)\nas well as other document types (for example, course materials).}\n\n\\section*{Acknowledgments}\n\nWe are grateful to Dr. C. Lee Giles for the CiteSeerX data. We also thank Corina Florescu and Kishore Neppalli for their help with various dataset construction tasks. This research was supported in part by the NSF award \\#1423337 to Cornelia Caragea. Any opinions, findings, and conclusions expressed here are those of the authors and do not necessarily reflect the views of NSF.\n\n\\small\n\\bibliographystyle{named}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}