We review the probabilistic properties of Ornstein-Uhlenbeck processes in Hilbert spaces driven by Levy processes. The emphasis is on the different contexts in which these processes arise, such as stochastic partial differential equations, continuous-state branching processes, generalised Mehler semigroups and operator self-decomposable distributions. We also examine generalisations to the case where the driving noise is cylindrical.
{"title":"Infinite dimensional Ornstein-Uhlenbeck processes driven by Lévy processes","authors":"D. Applebaum","doi":"10.1214/14-PS249","DOIUrl":"https://doi.org/10.1214/14-PS249","url":null,"abstract":"We review the probabilistic properties of Ornstein-Uhlenbeck \u0000processes in Hilbert spaces driven by Levy processes. The emphasis is on \u0000the different contexts in which these processes arise, such as stochastic partial \u0000differential equations, continuous-state branching processes, generalised \u0000Mehler semigroups and operator self-decomposable distributions. We also \u0000examine generalisations to the case where the driving noise is cylindrical.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"12 1","pages":"33-54"},"PeriodicalIF":1.6,"publicationDate":"2014-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/14-PS249","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66028288","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This survey paper is based on the lecture notes for the mini course in the summer school at Yau Mathematics Science Center, Tsinghua University, 2014. We describe and characterize all random subsets (K) of simply connected domain which satisfy the "conformal restriction" property. There are two different types of random sets: the chordal case and the radial case. In the chordal case, the random set (K) in the upper half-plane (mathbb{H}) connects two fixed boundary points, say 0 and (infty), and given that (K) stays in a simply connected open subset (H) of (mathbb{H}), the conditional law of (Phi(K)) is identical to that of (K), where (Phi) is any conformal map from (H) onto (mathbb{H}) fixing 0 and (infty ). In the radial case, the random set (K) in the upper half-plane (mathbb{H}) connects one fixed boundary points, say 0, and one fixed interior point, say (i), and given that (K) stays in a simply connected open subset (H) of (mathbb{H}), the conditional law of (Phi(K)) is identical to that of (K), where (Phi) is the conformal map from (H) onto (mathbb{H}) fixing 0 and (i). It turns out that the random set with conformal restriction property are closely related to the intersection exponents of Brownian motion. The construction of these random sets relies on Schramm Loewner Evolution with parameter (kappa=8/3) and Poisson point processes of Brownian excursions and Brownian loops.
{"title":"Conformal restriction and Brownian motion","authors":"Hao Wu","doi":"10.1214/15-PS259","DOIUrl":"https://doi.org/10.1214/15-PS259","url":null,"abstract":"This survey paper is based on the lecture notes for the mini course in \u0000the summer school at Yau Mathematics Science Center, Tsinghua \u0000University, 2014. \u0000 \u0000We describe and characterize all random subsets (K) of simply connected \u0000domain which satisfy the \"conformal restriction\" property. There are \u0000two different types of random sets: the chordal case and the radial \u0000case. In the chordal case, the random set (K) in the upper half-plane \u0000(mathbb{H}) connects two fixed boundary points, say 0 and (infty), and \u0000given that (K) stays in a simply connected open subset (H) of (mathbb{H}), \u0000the conditional law of (Phi(K)) is identical to that of (K), where \u0000(Phi) is any conformal map from (H) onto (mathbb{H}) fixing 0 and (infty \u0000). In the radial case, the random set (K) in the upper half-plane (mathbb{H}) \u0000connects one fixed boundary points, say 0, and one fixed interior \u0000point, say (i), and given that (K) stays in a simply connected open \u0000subset (H) of (mathbb{H}), the conditional law of (Phi(K)) is identical to \u0000that of (K), where (Phi) is the conformal map from (H) onto (mathbb{H}) \u0000fixing 0 and (i). \u0000 \u0000It turns out that the random set with conformal restriction property \u0000are closely related to the intersection exponents of Brownian motion. \u0000The construction of these random sets relies on Schramm Loewner \u0000Evolution with parameter (kappa=8/3) and Poisson point processes of \u0000Brownian excursions and Brownian loops. \u0000 \u0000","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"12 1","pages":"55-103"},"PeriodicalIF":1.6,"publicationDate":"2014-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/15-PS259","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66046677","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A. Lodhia, S. Sheffield, Xin Sun, Samuel S. Watson
We discuss a family of random fields indexed by a parameter s ∈ R which we call the fractional Gaussian fields, given by FGFs(R) = (−∆)−s/2W, where W is a white noise on Rd and (−∆)−s/2 is the fractional Laplacian. These fields can also be parameterized by their Hurst parameter H = s − d/2. In one dimension, examples of FGFs processes include Brownian motion (s = 1) and fractional Brownian motion (1/2 < s < 3/2). Examples in arbitrary dimension include white noise (s = 0), the Gaussian free field (s = 1), the bi-Laplacian Gaussian field (s = 2), the log-correlated Gaussian field (s = d/2), Levy’s Brownian motion (s = d/2 + 1/2), and multidimensional fractional Brownian motion (d/2 < s < d/2 + 1). These fields have applications to statistical physics, early-universe cosmology, finance, quantum field theory, image processing, and other disciplines. We present an overview of fractional Gaussian fields including covariance formulas, Gibbs properties, spherical coordinate decompositions, restrictions to linear subspaces, local set theorems, and other basic results. We also define a discrete fractional Gaussian field and explain how the FGFs with s ∈ (0, 1) can be understood as a long range Gaussian free field in which the potential theory of Brownian motion is replaced by that of an isotropic 2s-stable Levy process. ∗Partially supported by NSF grant DMS 1209044. †Supported by NSF GRFP award number 1122374. ar X iv :1 40 7. 55 98 v1 [ m at h. PR ] 2 1 Ju l 2 01 4
我们讨论了一类以参数s∈R为索引的随机场，我们称之为分数阶高斯场，由fgf (R) =(-∆)- s/2W给出，其中W是Rd上的白噪声，(-∆)- s/2是分数阶拉普拉斯函数。这些字段也可以通过它们的Hurst参数H = s−d/2来参数化。在一维中，FGFs过程的例子包括布朗运动(s = 1)和分数布朗运动(1/2 < s < 3/2)。任意维度的例子包括白噪声(s = 0)、高斯自由场(s = 1)、双拉普拉斯高斯场(s = 2)、对数相关高斯场(s = d/2)、利维布朗运动(s = d/2 + 1/2)和多维分数布朗运动(d/2 < s < d/2 + 1)。这些领域在统计物理、早期宇宙宇宙学、金融、量子场论、图像处理等学科中都有应用。我们概述了分数阶高斯场，包括协方差公式、吉布斯性质、球坐标分解、线性子空间的限制、局部集定理和其他基本结果。我们还定义了一个离散分数高斯场，并解释了如何将s∈(0,1)的fgf理解为一个远程高斯自由场，其中布朗运动的势理论被各向同性2s稳定Levy过程的势理论所取代。*部分由NSF资助DMS 1209044。†由NSF GRFP奖号1122374支持。ar X iv:1 40 7。55 98 v1 [m at h. PR] 2 1 Ju 1 2 01 4
{"title":"Fractional Gaussian fields: A survey","authors":"A. Lodhia, S. Sheffield, Xin Sun, Samuel S. Watson","doi":"10.1214/14-PS243","DOIUrl":"https://doi.org/10.1214/14-PS243","url":null,"abstract":"We discuss a family of random fields indexed by a parameter s ∈ R which we call the fractional Gaussian fields, given by FGFs(R) = (−∆)−s/2W, where W is a white noise on Rd and (−∆)−s/2 is the fractional Laplacian. These fields can also be parameterized by their Hurst parameter H = s − d/2. In one dimension, examples of FGFs processes include Brownian motion (s = 1) and fractional Brownian motion (1/2 < s < 3/2). Examples in arbitrary dimension include white noise (s = 0), the Gaussian free field (s = 1), the bi-Laplacian Gaussian field (s = 2), the log-correlated Gaussian field (s = d/2), Levy’s Brownian motion (s = d/2 + 1/2), and multidimensional fractional Brownian motion (d/2 < s < d/2 + 1). These fields have applications to statistical physics, early-universe cosmology, finance, quantum field theory, image processing, and other disciplines. We present an overview of fractional Gaussian fields including covariance formulas, Gibbs properties, spherical coordinate decompositions, restrictions to linear subspaces, local set theorems, and other basic results. We also define a discrete fractional Gaussian field and explain how the FGFs with s ∈ (0, 1) can be understood as a long range Gaussian free field in which the potential theory of Brownian motion is replaced by that of an isotropic 2s-stable Levy process. ∗Partially supported by NSF grant DMS 1209044. †Supported by NSF GRFP award number 1122374. ar X iv :1 40 7. 55 98 v1 [ m at h. PR ] 2 1 Ju l 2 01 4","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"13 1","pages":"1-56"},"PeriodicalIF":1.6,"publicationDate":"2014-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/14-PS243","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66027458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Several characterizations of the Generalized Inverse Gaussian (GIG) distribution on the positive real line have been proposed in the literature, especially over the past two decades. These characterization theorems are surveyed, and two new characterizations are established, one based on maximum likelihood estimation and the other is a Stein characterization.
{"title":"Characterizations of GIG laws: A survey","authors":"A. Koudou, Christophe Ley","doi":"10.1214/13-PS227","DOIUrl":"https://doi.org/10.1214/13-PS227","url":null,"abstract":"Several characterizations of the Generalized Inverse Gaussian (GIG) distribution on the positive real line have been proposed in the literature, especially over the past two decades. These characterization theorems are surveyed, and two new characterizations are established, one based on maximum likelihood estimation and the other is a Stein characterization.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"11 1","pages":"161-176"},"PeriodicalIF":1.6,"publicationDate":"2014-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66001914","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Localization and dilation procedures are discussed for infinite dimensional �-concave measures on abstract locally convex spaces (following Borell’s hierarchy of hyperbolic measures).
讨论了抽象局部凸空间上无限维凹测度的局部化和扩张过程(遵循Borell的双曲测度层次)。
{"title":"Hyperbolic measures on infinite dimensional spaces","authors":"S. Bobkov, J. Melbourne","doi":"10.1214/14-PS238","DOIUrl":"https://doi.org/10.1214/14-PS238","url":null,"abstract":"Localization and dilation procedures are discussed for infinite dimensional �-concave measures on abstract locally convex spaces (following Borell’s hierarchy of hyperbolic measures).","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"32 1","pages":"57-88"},"PeriodicalIF":1.6,"publicationDate":"2014-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66027833","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article we survey properties of mixed Poisson distributions and probabilistic aspects of the Stirling transform: given a non-negative random variable (X) with moment sequence ((mu_s)_{sinmathbb{N}}) we determine a discrete random variable (Y), whose moment sequence is given by the Stirling transform of the sequence ((mu_s)_{sinmathbb{N}}), and identify the distribution as a mixed Poisson distribution. We discuss properties of this family of distributions and present a new simple limit theorem based on expansions of factorial moments instead of power moments. Moreover, we present several examples of mixed Poisson distributions in the analysis of random discrete structures, unifying and extending earlier results. We also add several entirely new results: we analyse triangular urn models, where the initial configuration or the dimension of the urn is not fixed, but may depend on the discrete time (n). We discuss the branching structure of plane recursive trees and its relation to table sizes in the Chinese restaurant process. Furthermore, we discuss root isolation procedures in Cayley trees, a parameter in parking functions, zero contacts in lattice paths consisting of bridges, and a parameter related to cyclic points and trees in graphs of random mappings, all leading to mixed Poisson-Rayleigh distributions. Finally, we indicate how mixed Poisson distributions naturally arise in the critical composition scheme of Analytic Combinatorics.
{"title":"On moment sequences and mixed Poisson distributions","authors":"Markus Kuba, A. Panholzer","doi":"10.1214/14-PS244","DOIUrl":"https://doi.org/10.1214/14-PS244","url":null,"abstract":"In this article we survey properties of mixed Poisson distributions and \u0000probabilistic aspects of the Stirling transform: given a non-negative \u0000random variable (X) with moment sequence ((mu_s)_{sinmathbb{N}}) we \u0000determine a discrete random variable (Y), whose moment sequence is \u0000given by the Stirling transform of the sequence ((mu_s)_{sinmathbb{N}}), and \u0000identify the distribution as a mixed Poisson distribution. We discuss \u0000properties of this family of distributions and present a new simple \u0000limit theorem based on expansions of factorial moments instead of power \u0000moments. Moreover, we present several examples of mixed Poisson \u0000distributions in the analysis of random discrete structures, unifying \u0000and extending earlier results. We also add several entirely new \u0000results: we analyse triangular urn models, where the initial \u0000configuration or the dimension of the urn is not fixed, but may depend \u0000on the discrete time (n). We discuss the branching structure of plane \u0000recursive trees and its relation to table sizes in the Chinese \u0000restaurant process. Furthermore, we discuss root isolation procedures \u0000in Cayley trees, a parameter in parking functions, zero contacts in \u0000lattice paths consisting of bridges, and a parameter related to cyclic \u0000points and trees in graphs of random mappings, all leading to mixed \u0000Poisson-Rayleigh distributions. Finally, we indicate how mixed Poisson \u0000distributions naturally arise in the critical composition scheme of \u0000Analytic Combinatorics. \u0000 \u0000","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"13 1","pages":"89-155"},"PeriodicalIF":1.6,"publicationDate":"2014-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66027544","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Smoluchowski coagulation-diffusion PDE is a system of partial differential equations modelling the evolution in time of mass-bearing Brownian particles which are subject to short-range pairwise coagulation. This survey presents a fairly detailed exposition of the kinetic limit derivation of the Smoluchowski PDE from a microscopic model of many coagulating Brownian particles that was undertaken in [11]. It presents heuristic explanations of the form of the main theorem before discussing the proof, and presents key estimates in that proof using a novel probabilistic technique. The survey’s principal aim is an exposition of this kinetic limit derivation, but it also contains an overview of several topics which either motivate or are motivated by this derivation.
{"title":"Coagulation and diffusion: A probabilistic perspective on the Smoluchowski PDE","authors":"A. Hammond","doi":"10.1214/15-PS263","DOIUrl":"https://doi.org/10.1214/15-PS263","url":null,"abstract":"The Smoluchowski coagulation-diffusion PDE is a system of partial differential equations modelling the evolution in time of mass-bearing Brownian particles which are subject to short-range pairwise coagulation. This survey presents a fairly detailed exposition of the kinetic limit derivation of the Smoluchowski PDE from a microscopic model of many coagulating Brownian particles that was undertaken in [11]. It presents heuristic explanations of the form of the main theorem before discussing the proof, and presents key estimates in that proof using a novel probabilistic technique. The survey’s principal aim is an exposition of this kinetic limit derivation, but it also contains an overview of several topics which either motivate or are motivated by this derivation.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"14 1","pages":"205-288"},"PeriodicalIF":1.6,"publicationDate":"2014-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66047044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Consider a mathematical model with a finite number of random parameters. Variance based sensitivity analysis provides a framework to characterize the contribution of the individual parameters to the total variance of the model response. We consider the spectral methods for variance based sensitivity analysis which utilize representations of square integrable random variables in a generalized polynomial chaos basis. Taking a measure theoretic point of view, we provide a rigorous and at the same time intuitive perspective on the spectral methods for variance based sensitivity analysis. Moreover, we discuss approximation errors incurred by fixing inessential random parameters, when approximating functions with generalized polynomial chaos expansions.
{"title":"On spectral methods for variance based sensitivity analysis","authors":"A. Alexanderian","doi":"10.1214/13-PS219","DOIUrl":"https://doi.org/10.1214/13-PS219","url":null,"abstract":"Consider a mathematical model with a finite number of random parameters. \u0000Variance based sensitivity analysis provides a framework to characterize \u0000the contribution of the individual parameters to the total variance of \u0000the model response. We consider the spectral methods for variance based \u0000sensitivity analysis which \u0000utilize representations of square integrable random variables in a \u0000generalized polynomial chaos basis. \u0000Taking a measure theoretic point of view, we \u0000provide a rigorous and at the same time intuitive perspective on the \u0000spectral methods for variance based sensitivity analysis. \u0000Moreover, we discuss approximation errors incurred by fixing \u0000inessential random \u0000parameters, when approximating functions with generalized polynomial \u0000chaos expansions.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"10 1","pages":"51-68"},"PeriodicalIF":1.6,"publicationDate":"2013-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66001674","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Size bias occurs famously in waiting-time paradoxes, undesirably in sampling schemes, and unexpectedly in connection with Stein's method, tightness, analysis of the lognormal distribution, Skorohod embedding, infinite divisibility, and number theory. In this paper we review the basics and survey some of these unexpected connections.
{"title":"Size bias for one and all","authors":"R. Arratia, L. Goldstein, F. Kochman","doi":"10.1214/13-PS221","DOIUrl":"https://doi.org/10.1214/13-PS221","url":null,"abstract":"Size bias occurs famously in waiting-time paradoxes, undesirably in sampling schemes, and unexpectedly in connection with Stein's method, tightness, analysis of the lognormal distribution, Skorohod embedding, infinite divisibility, and number theory. In this paper we review the basics and survey some of these unexpected connections.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2013-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/13-PS221","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66001795","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we review the theory of Gaussian multiplicative chaos initially introduced by Kahane’s seminal work in 1985. Though this beautiful paper faded from memory until recently, it already contains ideas and results that are nowadays under active investigation, like the construction of the Liouville measure in 2d-Liouville quantum gravity or thick points of the Gaussian Free Field. Also, we mention important extensions and generalizations of this theory that have emerged ever since and discuss a whole family of applications, ranging from nance, through the Kolmogorov-Obukhov model of turbulence to 2d-Liouville quantum gravity. This review also includes new results like the convergence of discretized Liouville measures on isoradial graphs (thus including the triangle and square lattices) towards the continuous Liouville measures (in the subcritical and critical case) or multifractal analysis of the measures in all dimensions.
{"title":"Gaussian multiplicative chaos and applications: A review","authors":"Rémi Rhodes, V. Vargas","doi":"10.1214/13-PS218","DOIUrl":"https://doi.org/10.1214/13-PS218","url":null,"abstract":"In this article, we review the theory of Gaussian multiplicative chaos initially introduced by Kahane’s seminal work in 1985. Though this beautiful paper faded from memory until recently, it already contains ideas and results that are nowadays under active investigation, like the construction of the Liouville measure in 2d-Liouville quantum gravity or thick points of the Gaussian Free Field. Also, we mention important extensions and generalizations of this theory that have emerged ever since and discuss a whole family of applications, ranging from nance, through the Kolmogorov-Obukhov model of turbulence to 2d-Liouville quantum gravity. This review also includes new results like the convergence of discretized Liouville measures on isoradial graphs (thus including the triangle and square lattices) towards the continuous Liouville measures (in the subcritical and critical case) or multifractal analysis of the measures in all dimensions.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"31 1","pages":"315-392"},"PeriodicalIF":1.6,"publicationDate":"2013-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/13-PS218","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66001648","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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