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Stochastic Analysis and Related Topics

Part of the Progress in Probability Book Series (volume 72)

Editors: Baudoin, Fabrice; Peterson, Jonathon

DOI: 10.1007/978-3-319-59671-6

ISBN: 978-3-319-59670-9

Publisher: Springer International Publishing - Springer Journals

Sep 27, 2017

1 - 10 of 10 Chapters

LitStream

Chapter 1Positive-Homogeneous Operators, Heat Kernel Estimates and the Legendre-Fenchel Transform

Randles, Evan; Saloff-Coste, Laurent

Pages: 1 –55

[We consider a class of homogeneous partial differential operators on a finite-dimensional vector space and study their associated heat kernels. The heat kernels for this general class of operators are seen to arise naturally as the limiting objects of the convolution powers of complex-valued...
LitStream

Chapter 2Strong Stability of Heat Kernels of Non-symmetric Stable-Like Operators

Chen, Zhen-Qing; Zhang, Xicheng

Pages: 57 –65

[Let d≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$d\geqslant 1$$...
LitStream

Chapter 3Multiplicative Functional for the Heat Equation on Manifolds with Boundary

Ouyang, Cheng

Pages: 67 –83

[The multiplicative functional for the heat equation on k-forms with absolute boundary condition is constructed and a probabilistic representation of the solution is obtained. As an application, we prove a heat kernel domination that was previously discussed by Donnelly and Li, and Shigekawa.]
LitStream

Chapter 4Connections Between the Dirichlet and the Neumann Problem for Continuous and Integrable Boundary Data

Beznea, Lucian; Pascu, Mihai N.; Pascu, Nicolae R.

Pages: 85 –97

[We present results concerning the representation of the solution of the Neumann problem for the Laplace operator on the n-dimensional unit ball in terms of the solution of an associated Dirichlet problem. We show that the representation holds in the case of integrable boundary data, thus...
LitStream

Chapter 5Decomposition and Limit Theorems for a Class of Self-Similar Gaussian Processes

Harnett, Daniel; Nualart, David

Pages: 99 –116

[We introduce a new class of self-similar Gaussian stochastic processes, where the covariance is defined in terms of a fractional Brownian motion and another Gaussian process. A special case is the solution in time to the fractional-colored stochastic heat equation described in Tudor (Analysis of...
LitStream

Chapter 6On A Priori Estimates for Rough PDEs

Feng, Qi; Tindel, Samy

Pages: 117 –138

[In this note, we present a new and simple method which allows to get a priori bounds on rough partial differential equations. The technique is based on a weak formulation of the equation and a rough version of Gronwall’s lemma. The method is presented on a simple linear example, but might be...
LitStream

Chapter 7Lévy Systems and Moment Formulas for Mixed Poisson Integrals

Bogdan, Krzysztof; Rosiński, Jan; Serafin, Grzegorz; Wojciechowski, Łukasz

Pages: 139 –164

[We propose Mecke-Palm formula for multiple integrals with respect to the Poisson random measure and its intensity measure performed, or mixed, in an arbitrary order. We apply the formulas to mixed Lévy systems of Lévy processes and obtain moment formulas for mixed Poisson integrals.]
LitStream

Chapter 8Conformal Transforms and Doob’s h-Processes on Heisenberg Groups

Wang, Jing

Pages: 165 –177

[We study the stochastic processes that are images of Brownian motions on Heisenberg group H2n+1 under conformal maps. In particular, we obtain that Cayley transform maps Brownian paths in H2n+1 to a time changed Brownian motion on CR sphere 𝕊2n+1\documentclass[12pt]{minimal}...
LitStream

Chapter 9On the Macroscopic Fractal Geometry of Some Random Sets

Khoshnevisan, Davar; Xiao, Yimin

Pages: 179 –206

[This paper is concerned mainly with the macroscopic fractal behavior of various random sets that arise in modern and classical probability theory. Among other things, it is shown here that the macroscopic behavior of Boolean coverage processes is analogous to the microscopic structure of the...
LitStream

Chapter 10Critical Behavior of Mean-Field XY and Related Models

Kirkpatrick, Kay; Nawaz, Tayyab

Pages: 207 –221

[We discuss spin models on complete graphs in the mean-field (infinite-vertex) limit, especially the classical XY model, the Toy model of the Higgs sector, and related generalizations. We present a number of results coming from the theory of large deviations and Stein’s method, in particular,...

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