Stochastic Homogenization of Quasilinear PDEs with a Spatial Degeneracy

François Delarue (1) , Rémi Rhodes (2)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

We investigate stochastic homogenization for some degenerate quasilinear pa rabolic PDEs. The underlying nonlinear operator degenerates along the space variable, uniformly in the nonlinear term: the degeneracy points correspond to the degeneracy points of a reference diffusion operator on the random medium. Assuming that this reference diffusion operator is ergodic, we can prove the homogenization property for the quasilinear PDEs, by means of the first order approximation method. The (nonlinear) limit operator needn't be nondegenerate. Concrete examples are provided.

Keywords: ergodic operator; Stochastic homogenization; parabolic PDE; nonlinear PDE; degenerate PDE; first order approximation; ergodic operator.

Domains: Probability [math.PR]

François Delarue, Rémi Rhodes. Stochastic Homogenization of Quasilinear PDEs with a Spatial Degeneracy. Asymptotic Analysis, 2009, pp.61--90. ⟨hal-00141167⟩