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This paper is devoted to the study of the large time behaviour of viscositysolutions of parabolic equations with Neumann boundary conditions. This work isthe sequel of [13] in which a probabilistic method was developped to show thatthe solution of a parabolic semilinear PDE behaves like a linear term $\lambdaT$ shifted with a function $v$, where $(v,\lambda)$ is the solution of theergodic PDE associated to the parabolic PDE. We adapt this method in finitedimension by a penalization method in order to be able to apply an importantbasic coupling estimate result and with the help of a regularization procedurein order to avoid the lack of regularity of the coefficients in finitedimension. The advantage of our method is that it gives an explicit rate ofconvergence.

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