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Existence of global martingale solutions to stochastic hyperbolic equations driven by a spatially homogeneous Wiener process. (English) Zbl 1092.60024
The author considers a stochastic partial differential equation \[ u_{tt}={\mathcal A}u+f(u)+g(u)\dot W,\quad u(0)=u_0,\;u_t(0)=v_0, \] in a domain \(G\subseteq {\mathbb R}^d\), where \(\mathcal A\) is a uniformly elliptic second-order differential operator, \(f,g\) are real continuous non-Lipschitz functions and \(W\) is a spatially homogeneous Wiener process in the tempered distribution space \({\mathcal S}'({\mathbb R}^d)\) with finite spectral measure. \(D\) can be either whole \({\mathbb R}^d\) or a bounded set with \(C^2\)-boundary, and then either Dirichlet or Neumann conditions are imposed. A solution of this equation is understood as a weak (i.e., defined via a suitable duality) solution of a martingale problem, appropriately (i.e., in a rather non-standard way) defined. It is proved that under some, quite intricate, growth conditions on \(f\) and \(g\), a global solution exists, whereas uniqueness is not discussed.

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