The Non-Linear Stochastic Wave Equation in High Dimensions

Daniel Conus (Ecole Polytechnique Fédérale)
Robert C. Dalang (Ecole Polytechnique Fédérale)

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 ($p\geq 1$), and we show that the solution is Hölder continuous. The Hölder exponent that we obtain is optimal.

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Pages: 629-670

Publication Date: April 12, 2008

DOI: 10.1214/EJP.v13-500

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