The non-linear stochastic wave equation in high dimensions. (English) Zbl 1187.60049

Conus, Daniel; Dalang, Robert C.

Electron. J. Probab. 13, 629-670 (2008).

Summary: 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.

MSC:

60H15

Stochastic partial differential equations (aspects of stochastic analysis)

Keywords:

martingale measures; stochastic integration; stochastic wave equation; stochastic partial differential equations; moment formulae; Hölder continuity