Stochastic geometric wave equations with values in compact Riemannian homogeneous spaces

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

2011-01-25

arxiv.org/abs/1101.4835v1

Mathematics

Probability

Scientific paper

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.

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