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Weak solutions to stochastic wave equations with values in Riemannian manifolds. (English) Zbl 1238.60073
Authors’ abstract: Let \(M\) be a compact Riemannian manifold. We prove existence of a global weak solution of the stochastic wave equation \(D_tu_t=D_xu_x+(X_u+\lambda_{0}(u)u_t+\lambda_{1}(u)u_x)\dot{W}\), where \(X\) is a continuous vector field on \(M\), \(\lambda_{0}\) and \(\lambda_{1}\) are continuous vector bundle homomorphisms from \(TM\) to \(TM\), and \(W\) is a spatially homogeneous Wiener process on \(\mathbb{R}\) with finite spectral measure. We use recently introduced general method of constructing weak solutions of SPDEs that does not rely on any martingale representation theorem.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
58J65 Diffusion processes and stochastic analysis on manifolds
58E20 Harmonic maps, etc.
35L70 Second-order nonlinear hyperbolic equations
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