A probabilistic approach to large time behaviour of parabolic equations with Neumann boundary conditions

Ying Hu, Pierre-Yves Madec

Published 2015-03-26Version 1

This paper is devoted to the study of the large time behaviour of viscosity solutions of parabolic equations with Neumann boundary conditions. This work is the sequel of [12] in which a probabilistic method was developped to show that the solution of a parabolic semilinear PDE behaves like a linear term $\lambda$T shifted with a function v, where (v, $\lambda$) is the solution of the ergodic PDE associated to the parabolic PDE. We adapt this method in finite dimension by a penalization method in order to be able to apply an important basic coupling estimate result and with the help of a regularization procedure in order to avoid the lack of regularity of the coefficients in finite dimension.