Stochastic hyperbolic systems and the wave equation. (English) Zbl 0739.60059

Summary: We study initial-boundary value problems for certain systems of first order stochastic hyperbolic equations in a quarter plane \(\mathbb{R}^ 2_ +\) or in a strip \(\mathbb{R}_ +\times(0,A)\), with various types of boundary conditions, including nonlocal conditions. We propose a framework for the study of such problems and show that under appropriate conditions the problems are well posed namely, there exists a unique solution which yields a strong Markov process. Following these general results we discuss several examples of initial-boundary value problems for the stochastic wave equation. Concerning these problems we observe that the regularity of the solutions is strongly affected by the reflection of the ‘noise’ at the boundary. This effect depends on the specific nature of the boundary conditions.


60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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