Stochastic geometric wave equations with values in compact Riemannian homogeneous spaces

Abstract

Let $M$ be a compact Riemannian homogeneous space (e.g., a Euclidean sphere). We prove existence of a global weak solution of the stochastic wave equation $\mathbf{D}_{t}\partial_{t}u=\sum_{k=1}^{d}\mathbf{D}_{x_{k}}\partial_{x_{k}}u+f_{u}(Du)+g_{u}(Du)\dot{W}$ in any dimension $d\ge1$, where $f$ and $g$ are continuous multilinear maps, and $W$ is a spatially homogeneous Wiener process on $\mathbb{R}^{d}$ with finite spectral measure. A nonstandard method of constructing weak solutions of SPDEs, that does not rely on martingale representation theorem, is employed.