# Stochastic geometric wave equations with values in compact Riemannian homogeneous spaces

Zdzislaw Brzezniak, Martin Ondreját

Research output: Contribution to journalArticlepeer-review

## 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.
Original language English 1938-1977 44 Annals of Probability 41 3B https://doi.org/10.1214/11-AOP690 Published - May 2013

## Keywords

• stochastic wave equation,
• Riemannian manifold,
• homogeneous space.
• ### Some questions related to invariant measures for stochastic Navier Stokes equations

Brzezniak, Z.

EPSRC

29/11/0628/04/09

Project: Research project (funded)Research