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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.

##### MSC:
 60H15 Stochastic partial differential equations (aspects of stochastic analysis)
##### Keywords:
stochastic wave equation
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##### References:
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