The non-linear stochastic wave equation in high dimensions

Abstract

We propose an extension of Walsh’s classical martingale measure stochastic integral that makes it possible to integrate a general class of Schwartz distributions, which contains the fundamental solution of the wave equation, even in dimensions greater than 3. This leads to a square-integrable random-field solution to the non-linear stochastic wave equation in any dimension, in the case of a driving noise that is white in time and correlated in space. In the particular case of an affine multiplicative noise, we obtain estimates on p-th moments of the solution (p ≣ 1), and we show that the solution is Hölder continuous. The Hölder exponent that we obtain is optimal . © 2008 Applied Probability Trust.