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

by Robert C. Dalang, Marta Sanz-Sole
     
 

ISBN-10: 0821842889

ISBN-13: 9780821842881

Pub. Date: 04/10/2009

Publisher: American Mathematical Society

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

Overview

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.

Product Details

ISBN-13:
9780821842881
Publisher:
American Mathematical Society
Publication date:
04/10/2009
Series:
Memoirs of the American Mathematical Society Series, #199
Pages:
70

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