Ergodic BSDEs under weak dissipative assumptions

In this paper we study ergodic backward stochastic differential equations (EBSDEs) dropping the strong dissipativity assumption needed in Fuhrman et al. (2009) [12]. In other words we do not need to require the uniform exponential decay of the difference of two solutions of the underlying forward equation, which, on the contrary, is assumed to be non-degenerate. We show the existence of solutions by the use of coupling estimates for a non-degenerate forward stochastic differential equation with bounded measurable nonlinearity. Moreover we prove the uniqueness of "Markovian" solutions by exploiting the recurrence of the same class of forward equations. Applications are then given for the optimal ergodic control of stochastic partial differential equations and to the associated ergodic Hamilton-Jacobi-Bellman equations.Backward stochastic differential equation Bismut-Elworthy formula Coupling estimate Ergodic control Hamilton-Jacobi-Bellman equation Recurrence property Weak dissipative assumption