{"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}
{"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
{"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}
{"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}
{"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}
{"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}
{"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}
{"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}
{"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}
{"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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