Stochastic geometric wave equations with values in compact Riemannian homogeneous spaces

TY - JOUR

T1 - Stochastic geometric wave equations with values in compact Riemannian homogeneous spaces

AU - Brzezniak, Zdzislaw

AU - Ondreját, Martin

PY - 2013/5

Y1 - 2013/5

N2 - 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.

AB - 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.

KW - stochastic wave equation,

KW - Riemannian manifold,

KW - homogeneous space.

U2 - 10.1214/11-AOP690

DO - 10.1214/11-AOP690

M3 - Article

SN - 0091-1798

VL - 41

SP - 1938

EP - 1977

JO - Annals of Probability

JF - Annals of Probability

IS - 3B

ER -