Weak solutions to stochastic wave equations with values in Riemannian manifolds

Zdzislaw Brzezniak; Martin Ondreját

doi:10.1080/03605302.2011.574243

Weak solutions to stochastic wave equations with values in Riemannian manifolds

Zdzislaw Brzezniak, Martin Ondreját

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let M be a compact Riemannian manifold. We prove existence of a global
weak solution of the stochastic wave equation Dt@tu = Dx@xu + (Xu + ¸0(u)@tu +
¸1(u)@xu) ¿W where X is a continuous tangent vector field on M, ¸0, ¸1 are continuous vector bundles homomorphisms from TM to TM and W is a spatially homogeneous Wiener process on R with finite spectral measure. A new general method of constructing weak solutions of SPDEs that does not rely on martingale representation theorem is used.

Original language	English
Pages (from-to)	1624-1653
Number of pages	30
Journal	Communications in Partial Differential Equations
Volume	36
Issue number	9
DOIs	https://doi.org/10.1080/03605302.2011.574243
Publication status	Published - Sept 2011

Keywords

stochastic wave equation;
geometric wave equation.

Access to Document

10.1080/03605302.2011.574243

http://www.tandfonline.com/doi/abs/10.1080/03605302.2011.574243

Cite this

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title = "Weak solutions to stochastic wave equations with values in Riemannian manifolds",

abstract = "Let M be a compact Riemannian manifold. We prove existence of a global weak solution of the stochastic wave equation Dt@tu = Dx@xu + (Xu + ¸0(u)@tu + ¸1(u)@xu) ¿W where X is a continuous tangent vector field on M, ¸0, ¸1 are continuous vector bundles homomorphisms from TM to TM and W is a spatially homogeneous Wiener process on R with finite spectral measure. A new general method of constructing weak solutions of SPDEs that does not rely on martingale representation theorem is used.",

keywords = "stochastic wave equation; , geometric wave equation.",

author = "Zdzislaw Brzezniak and Martin Ondrej{\'a}t",

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AU - Ondreját, Martin

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AB - Let M be a compact Riemannian manifold. We prove existence of a global weak solution of the stochastic wave equation Dt@tu = Dx@xu + (Xu + ¸0(u)@tu + ¸1(u)@xu) ¿W where X is a continuous tangent vector field on M, ¸0, ¸1 are continuous vector bundles homomorphisms from TM to TM and W is a spatially homogeneous Wiener process on R with finite spectral measure. A new general method of constructing weak solutions of SPDEs that does not rely on martingale representation theorem is used.

KW - stochastic wave equation;

KW - geometric wave equation.

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DO - 10.1080/03605302.2011.574243

M3 - Article

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