Probabilistic Interpretation of a System of Quasilinear Parabolic Pdes

Auxiliary Sdes for Homogenization of Quasilinear Pdes with Periodic Coefficients

We study the homogenization property of systems of quasi-linear PDEs of parabolic type with periodic coefficients, highly oscillating drift and highly oscillating nonlinear term. To this end, we propose a probabilistic approach based on the theory of forward–backward stochastic differential equations and introduce the new concept of " auxiliary SDEs. " 1. Introduction and assumptions.

Backward doubly stochastic differential equations and systems of quasilinear SPDEs

A new kind of backward stochastic differential equations (in short BSDE), where the solution is a pair of processes adapted to the past of the driving Brownian motion, has been introduced by the authors in [63. It was then shown in a series of papers by the second and both authors (see [8, 7, 9, 103), that this kind of backward SDEs gives a probabilistic representation for the solution of a lar...

Maximum principle for quasilinear stochastic PDEs with obstacle

We prove a maximum principle for local solutions of quasilinear stochastic PDEs with obstacle (in short OSPDE). The proofs are based on a version of Itô’s formula and estimates for the positive part of a local solution which is non-positive on the lateral boundary. Our method is based on a version of Moser’s iteration scheme developed first by Aronson and Serrin [2] in the context of non-linear...

A Quasilinear Parabolic Equation With Inhomogeneous Density and Absorption

Stochastic homogenization of quasilinear PDEs with a spatial degeneracy

We investigate stochastic homogenization for some degenerate quasilinear parabolic 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 homogenizati...