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Article Dans Une Revue Annals of Probability Année : 2016

Degenerate Parabolic Stochastic Partial Differential Equations: Quasilinear case

Résumé

We study the Cauchy problem for a quasilinear degenerate parabolic stochastic partial differential equation driven by a cylindrical Wiener process. In particular, we adapt the notion of kinetic formulation and kinetic solution and develop a well-posedness theory that includes also an $L^1$-contraction property. In comparison to the first-order case (Debussche and Vovelle, 2010) and to the semilinear degenerate parabolic case (Hofmanová, 2013), the present result contains two new ingredients: a generalized Itô formula that permits a rigorous derivation of the kinetic formulation even in the case of weak solutions of certain nondegenerate approximations and a direct proof of strong convergence of these approximations to the desired kinetic solution of the degenerate problem.
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Dates et versions

hal-00863829 , version 1 (19-09-2013)
hal-00863829 , version 2 (20-11-2018)

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Arnaud Debussche, Martina Hofmanova, Julien Vovelle. Degenerate Parabolic Stochastic Partial Differential Equations: Quasilinear case. Annals of Probability, 2016, 44 (3), pp.1916-1955. ⟨10.1214/15-AOP1013⟩. ⟨hal-00863829v2⟩
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