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

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*Robert C. Dalang; Marta Sanz-Solé*

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.

#### Table of Contents

# Table of Contents

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

- Contents v6 free
- Chapter 1. Introduction 18 free
- Chapter 2. The Fundamental Solution of the Wave Equation and the Covariance Function 714 free
- Chapter 3. Hölder-Sobolev Regularity of the Stochastic Integral 1320
- Chapter 4. Path Properties of the Solution of the Stochastic Wave Equation 3340
- Chapter 5. Sharpness of the Results 4956
- Chapter 6. Integrated Increments of the Covariance Function 5360
- Bibliography 6976