The Annals of Probability

Degenerate parabolic stochastic partial differential equations: Quasilinear case

Arnaud Debussche, Martina Hofmanová, and Julien Vovelle

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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.

Article information

Ann. Probab., Volume 44, Number 3 (2016), 1916-1955.

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

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Zentralblatt MATH identifier

Primary: 60H15: Stochastic partial differential equations [See also 35R60] 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15]

Quasilinear degenerate parabolic stochastic partial differential equation kinetic formulation kinetic solution


Debussche, Arnaud; Hofmanová, Martina; Vovelle, Julien. Degenerate parabolic stochastic partial differential equations: Quasilinear case. Ann. Probab. 44 (2016), no. 3, 1916--1955. doi:10.1214/15-AOP1013.

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