Holder-Sobolev Regularity of the Solution to the Stochastic Wave Equation in Dimension Three
Holder-Sobolev Regularity of the Solution to the Stochastic Wave Equation in Dimension Three

Author: Robert C. Dalang

Publisher: American Mathematical Soc.

Published: 2009-04-10

Total Pages: 83

ISBN-13: 0821842889

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The authors study the sample path regularity of the solution of a stochastic wave equation in spatial dimension $d=3$. The driving noise is white in time and with a spatially homogeneous covariance defined as a product of a Riesz kernel and a smooth function. The authors prove that at any fixed time, a.s., the sample paths in the spatial variable belong to certain fractional Sobolev spaces. In addition, for any fixed $x\in\mathbb{R}^3$, the sample paths in time are Holder continuous functions. Further, the authors obtain joint Holder continuity in the time and space variables. Their results rely on a detailed analysis of properties of the stochastic integral used in the rigourous formulation of the s.p.d.e., as introduced by Dalang and Mueller (2003). Sharp results on one- and two-dimensional space and time increments of generalized Riesz potentials are a crucial ingredient in the analysis of the problem. For spatial covariances given by Riesz kernels, the authors show that the Holder exponents that they obtain are optimal.

General Stochastic Measures
General Stochastic Measures

Author: Vadym M. Radchenko

Publisher: John Wiley & Sons

Published: 2022-09-21

Total Pages: 276

ISBN-13: 1786308282

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This book is devoted to the study of stochastic measures (SMs). An SM is a sigma-additive in probability random function, defined on a sigma-algebra of sets. SMs can be generated by the increments of random processes from many important classes such as square-integrable martingales and fractional Brownian motion, as well as alpha-stable processes. SMs include many well-known stochastic integrators as partial cases. General Stochastic Measures provides a comprehensive theoretical overview of SMs, including the basic properties of the integrals of real functions with respect to SMs. A number of results concerning the Besov regularity of SMs are presented, along with equations driven by SMs, types of solution approximation and the averaging principle. Integrals in the Hilbert space and symmetric integrals of random functions are also addressed. The results from this book are applicable to a wide range of stochastic processes, making it a useful reference text for researchers and postgraduate or postdoctoral students who specialize in stochastic analysis.

Level Sets and Stochastic Partial Differential Equations
Level Sets and Stochastic Partial Differential Equations

Author:

Publisher:

Published: 1996

Total Pages: 7

ISBN-13:

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The effort reported on here was primarily aimed at acquiring a better understanding of a broad class of stochastic partial differential equations. The main class of problems was concerned with regularity properties of solutions to stochastic wave equations in one and two spatial dimensions. A second class of problems arose from attempts to understand the flow of information throughout the solution of a linear stochastic wave equation in two spatial dimensions driven by Levy (shock) noise. A third topic studied was in the area of stochastic optimization. Substantial results have been obtained in all three areas. These results have given rise to six published (or soon to be published) research articles, a published monograph and a Ph. D. thesis.

A Minicourse on Stochastic Partial Differential Equations
A Minicourse on Stochastic Partial Differential Equations

Author: Robert C. Dalang

Publisher: Springer Science & Business Media

Published: 2009

Total Pages: 230

ISBN-13: 3540859934

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This title contains lectures that offer an introduction to modern topics in stochastic partial differential equations and bring together experts whose research is centered on the interface between Gaussian analysis, stochastic analysis, and stochastic PDEs.