We study the large time behaviour of mild solutions of HJB equations in infinite dimension by a purely probabilistic approach. For that purpose, we show that the solution of a BSDE in finite horizon T taken at initial time behaves like a linear term in T shifted with the solution of the associated EBSDE taken at initial time. Moreover we give an explicit rate of convergence, which seems to be new up to our best knowledge.Key words. Backward stochastic differential equations; Ergodic backward stochastic differential equations; HJB equations in infinite dimension; Large time behaviour; Mild solutions; Ornstein-Uhlenbeck operator.

We study a class of ergodic BSDEs related to PDEs with Neumann boundary conditions. The randomness of the driver is given by a forward process under weakly dissipative assumptions with an invertible and bounded diffusion matrix. Furthermore, this forward process is reflected in a convex subset of R d not necessarily bounded. We study the link of such EBSDEs with PDEs and we apply our results to an ergodic optimal control problem.

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 developed to show that the solution of a parabolic semilinear PDE behaves like a linear term λT shifted with a function v, where (v, λ) 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.