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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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[1] Aronson, D. G. (1963). On the Green’s function for second order parabolic differential equations with discontinous coefficients. Bull. Amer. Math. Soc. 69 841-847. · Zbl 0154.11903 · doi:10.1090/S0002-9904-1963-11059-9
[2] Aronson, D. G. (1968). Non-negative solutions of linear parabolic equations. Ann. Scuola Norm. Sup. Pisa (3) 22 607-694. · Zbl 0182.13802 · numdam:ASNSP_1968_3_22_4_607_0 · eudml:83474
[3] Bally, V., Caballero, E., El-Karoui, N. and Fernandez, B. (2004). Reflected BSDE’s PDE’s and variational inequalities. INRIA report.
[4] Bensoussan, A. and Lions, J. L. (1978). Applications des inéquations variationnelles en contrôle stochastique . Dunod, Paris. · Zbl 0411.49002
[5] Charrier, P. and Troianiello, G. M. (1975). Un résultat d’existence et de régularité pour les solutions fortes d’un problème unilatéral d’évolution avec obstacle dépendant du temps. C. R. Acad. Sci. Paris Sér. A-B 281 Aii, A621-A623. · Zbl 0321.35042
[6] Denis, L. (2004). Solutions of stochastic partial differential equations considered as Dirichlet processes. Bernoulli 10 783-827. · Zbl 1071.60054 · doi:10.3150/bj/1099579156
[7] Denis, L., Matoussi, A. and Stoïca, L. (2005). \(L^{p}\) estimates for the uniform norm of solutions of quasilinear SPDE. Probab. Theory Related Fields 133 437-463. · Zbl 1085.60043 · doi:10.1007/s00440-005-0436-5
[8] Denis, L. and Stoïca, L. (2004). A general analytical result for non-linear s.p.d.e.’s and applications. Electron. J. Probab. 9 674-709. · Zbl 1067.60048 · doi:10.1214/EJP.v9-223 · emis:journals/EJP-ECP/_ejpecp/EjpVol9/paper23.abs.html · eudml:124919
[9] Donati-Martin, C. and Pardoux, É. (1993). White noise driven SPDEs with reflection. Probab. Theory Related Fields 95 1-24. · Zbl 0794.60059 · doi:10.1007/BF01197335
[10] El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S. and Quenez, M. C. (1997). Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s. Ann. Probab. 25 702-737. · Zbl 0899.60047 · doi:10.1214/aop/1024404416
[11] Klimsiak, T. (2012). Reflected BSDEs and the obstacle problem for semilinear PDEs in divergence form. Stochastic Process. Appl. 122 134-169. · Zbl 1242.60070 · doi:10.1016/j.spa.2011.10.001
[12] Lions, J. L. and Magenes, E. (1968). Problèmes aux limites non homogènes et applications . Dunod, Paris. · Zbl 0165.10801
[13] Matoussi, A. and Stoïca, L. (2010). The obstacle problem for quasilinear stochastic PDE’s. Ann. Probab. 38 1143-1179. · Zbl 1200.60052 · doi:10.1214/09-AOP507
[14] Matoussi, A. and Xu, M. (2008). Sobolev solution for semilinear PDE with obstacle under monotonicity condition. Electron. J. Probab. 13 1035-1067. · Zbl 1191.35133 · doi:10.1214/EJP.v13-522 · emis:journals/EJP-ECP/_ejpecp/viewarticle97da.html · eudml:233051
[15] Mignot, F. and Puel, J. P. (1977). Inéquations d’évolution paraboliques avec convexes dépendant du temps. Applications aux inéquations quasi variationnelles d’évolution. Arch. Ration. Mech. Anal. 64 59-91. · Zbl 0362.49011 · doi:10.1007/BF00280179
[16] Nualart, D. and Pardoux, É. (1992). White noise driven quasilinear SPDEs with reflection. Probab. Theory Related Fields 93 77-89. · Zbl 0767.60055 · doi:10.1007/BF01195389
[17] Pierre, M. (1979). Problèmes d’evolution avec contraintes unilatérales et potentiels paraboliques. Comm. Partial Differential Equations 4 1149-1197. · Zbl 0426.31005 · doi:10.1080/03605307908820124
[18] Pierre, M. (1980). Représentant précis d’un potentiel parabolique. In Seminar on Potential Theory , Paris , No. 5 ( French ). Lecture Notes in Math. 814 186-228. Springer, Berlin. · Zbl 0463.31007
[19] Riesz, F. and Sz.-Nagy, B. (1990). Functional Analysis . Dover, New York. · Zbl 0732.47001
[20] Sanz-Solé, M. and Vuillermot, P.-A. (2003). Equivalence and Hölder-Sobolev regularity of solutions for a class of non-autonomous stochastic partial differential equations. Ann. Inst. Henri Poincaré Probab. Stat. 39 703-742. · Zbl 1026.60080 · doi:10.1016/S0246-0203(03)00015-3 · numdam:AIHPB_2003__39_4_703_0 · eudml:77778
[21] Xu, T. and Zhang, T. (2009). White noise driven SPDEs with reflection: Existence, uniqueness and large deviation principles. Stochastic Process. Appl. 119 3453-3470. · Zbl 1175.60068 · doi:10.1016/j.spa.2009.06.005
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