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

TL;DR

This paper develops a probabilistic method to analyze the large time behavior of viscosity solutions for parabolic equations with Neumann boundary conditions, providing explicit convergence rates.

Contribution

It adapts a probabilistic approach with penalization and regularization techniques to finite dimensions, enabling explicit convergence rate estimates.

Findings

Established a probabilistic framework for large time behavior.

Derived explicit convergence rates for viscosity solutions.

Extended methods to handle finite-dimensional PDEs with boundary conditions.

Abstract

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 [13] 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. The advantage of our method is that it gives an explicit rate of convergence.