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Strong solutions to stochastic wave equations with values in Riemannian manifolds. (English) Zbl 1141.58019
The authors study the stochastic wave equation with values in Riemannian manifolds (opposite to Euclidean spaces). On a $$d$$-dimensional compact manifold $$M$$, they consider the following one-dimensional stochastic wave equation $\mathbf{D}_t \partial_t u = \mathbf{D}_x \partial_x u + Y_u\, (\partial_t u, \partial_x u) \dot{W},$ where $$\mathbf{D}$$ is the connection on the pull-back bundle $$u^{-1} \text{TM}$$ induced by the Riemannian connection on $$M$$. They prove, after giving a rigorous formulation of the stochastic problem, existence and uniqueness of solutions of the above equation which are strong both in the PDE sense as well as in the probabilistic sense.

MSC:
 58J65 Diffusion processes and stochastic analysis on manifolds 60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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References:
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