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May 2013 Stochastic geometric wave equations with values in compact Riemannian homogeneous spaces
Zdzisław Brzeźniak, Martin Ondreját
Ann. Probab. 41(3B): 1938-1977 (May 2013). DOI: 10.1214/11-AOP690

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.

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Zdzisław Brzeźniak. Martin Ondreját. "Stochastic geometric wave equations with values in compact Riemannian homogeneous spaces." Ann. Probab. 41 (3B) 1938 - 1977, May 2013. https://doi.org/10.1214/11-AOP690

Information

Published: May 2013
First available in Project Euclid: 15 May 2013

zbMATH: 1286.60058
MathSciNet: MR3098063
Digital Object Identifier: 10.1214/11-AOP690

Subjects:
Primary: 60H15
Secondary: 35L05, 35R60, 58J65

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 3B • May 2013
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