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The obstacle problem for quasilinear stochastic PDEs: analytical approach. (English) Zbl 1298.60064
The authors consider a parabolic partial differential equation perturbed by an infinite number of independent Brownian motions $du=\partial_i(a_{ij}\partial_ju+g_i(u,\nabla u))\,dt+f(u,\nabla u)\,dt+h_j(u,\nabla u)\,dB_j+d\nu_t$ on a domain $$\mathcal O$$ in $$\mathbb R^d$$, with an initial condition $$u(0,x)=\xi(x)$$, with a homogeneous Dirichlet boundary condition, and with the obstacle condition $$u\geq S$$, where $$S$$ is a random function on $$\mathbb R_+\times\mathcal O$$. Here, $$(a_{ij})$$ is a uniformly elliptic symmetric bounded measurable matrix, $$\xi$$ is an $$L^2(\mathcal O)$$-valued random variable, $$g_i$$, $$f$$, $$h_i$$ are predictable Lipschitz functions on $$\mathbb R_+\times\Omega\times\mathcal O\times\mathbb R\times\mathbb R^d$$. It is proved that, provided $$S$$ is quasi-continuous (in terms of parabolic capacity) and verifies $$S\leq S^\prime$$, where $$S^\prime$$ is a solution to a certain linear stochastic partial differential equation and the nonlinearities $$g_i$$, $$f$$, $$h_i$$ satisfy an integrability condition, then the stochastic obstacle problem has a unique solution $$(u,\nu)$$, where $$u$$ is a predictable continuous process, $$\nu$$ is a random regular measure and $$u$$ has a quasi-continuous version $$\tilde u$$ that satisfies the minimal Skorokhod condition $$\langle\tilde u-S,\nu\rangle=0$$ a.s. Finally, the authors prove a comparison theorem for solutions of the stochastic obstacle problem.

##### MSC:
 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 31B15 Potentials and capacities, extremal length and related notions in higher dimensions
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##### References:
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