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Hölder-Sobolev Regularity of the Solution to the Stochastic Wave Equation in Dimension Three
Robert C. Dalang, Ecole Polytechnique Fédérale, Lausanne, Switzerland, and Marta Sanz-Solé, Universitat de Barcelona, Spain
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Memoirs of the American Mathematical Society
2009; 70 pp; softcover
Volume: 199
ISBN-10: 0-8218-4288-9
ISBN-13: 978-0-8218-4288-1
List Price: US$60 Individual Members: US$36
Institutional Members: US\$48
Order Code: MEMO/199/931

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 Hölder continuous functions. Further, the authors obtain joint Hölder 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 Hölder exponents that they obtain are optimal.