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.
Daniel Conus. Robert Dalang. "The Non-Linear Stochastic Wave Equation in High Dimensions." Electron. J. Probab. 13 629 - 670, 2008. https://doi.org/10.1214/EJP.v13-500