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May 2016 Degenerate parabolic stochastic partial differential equations: Quasilinear case
Arnaud Debussche, Martina Hofmanová, Julien Vovelle
Ann. Probab. 44(3): 1916-1955 (May 2016). DOI: 10.1214/15-AOP1013

Abstract

In this paper, 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 previous works of the authors concerning stochastic hyperbolic conservation laws [J. Funct. Anal. 259 (2010) 1014–1042] and semilinear degenerate parabolic SPDEs [Stochastic Process. Appl. 123 (2013) 4294–4336], the present result contains two new ingredients that provide simpler and more effective method of the proof: 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.

Citation

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Arnaud Debussche. Martina Hofmanová. Julien Vovelle. "Degenerate parabolic stochastic partial differential equations: Quasilinear case." Ann. Probab. 44 (3) 1916 - 1955, May 2016. https://doi.org/10.1214/15-AOP1013

Information

Received: 1 August 2013; Revised: 1 November 2014; Published: May 2016
First available in Project Euclid: 16 May 2016

zbMATH: 1346.60094
MathSciNet: MR3502597
Digital Object Identifier: 10.1214/15-AOP1013

Subjects:
Primary: 35R60, 60H15

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.44 • No. 3 • May 2016
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