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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.

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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