Stochastic geometric wave equations with values in compact Riemannian homogeneous spaces

TL;DR

This paper establishes the existence of global weak solutions for stochastic wave equations on compact Riemannian homogeneous spaces, using a novel construction method that avoids traditional martingale representation techniques.

Contribution

It introduces a new approach for constructing weak solutions to SPDEs on homogeneous spaces, applicable in any spatial dimension, without relying on martingale representation.

Findings

Proves existence of solutions in all dimensions $d\,\geq 1$

Develops a nonstandard solution construction method

Handles stochastic wave equations with spatially homogeneous noise

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_tu=\sum_{k=1}^d\mathbf D_{x_k}\partial_{x_k}u+f_u(Du)+g_u(Du)\,\dot W$ in any dimension $d\ge 1$, where $f$ and $g$ are continuous multilinear mappings 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.