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

Ying Hu (1) , Pierre-Yves Madec (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

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.

Mots clés: Viscosity solutions.; Backward stochastic differential equations; Ergodic backward stochastic differential equations; HJB equations; Large time bahaviour

Domaines: Probabilités [math.PR]

Ying Hu, Pierre-Yves Madec. A probabilistic approach to large time behaviour of viscosity solutions of parabolic equations with Neumann boundary conditions. Applied Mathematics and Optimization, 2016, 74 (2), pp.345-374. ⟨10.1007/s00245-015-9318-0⟩. ⟨hal-01135840v3⟩