Home » HOlder-Sobolev Regularity of the Solution to the Stochastic Wave Equation in Dimension Three by Robert C. Dalang
HOlder-Sobolev Regularity of the Solution to the Stochastic Wave Equation in Dimension Three Robert C. Dalang

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

Robert C. Dalang

Published January 1st 2009
ISBN : 9780821842881
Hardcover
70 pages
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 About the Book 

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 smoothMoreThe 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.