Ergodic BSDEs and optimal ergodic control in Banach spaces

In this paper we introduce a new kind of backward stochastic differential equations, called ergodic BSDEs, which arise naturally in the study of optimal ergodic control. We study the existence, uniqueness, and regularity of solution to ergodic BSDEs. Then we apply these results to the optimal ergodic control of a Banach valued stochastic state equation. We also establish the link between the ergodic BSDEs and the associated Hamilton–Jacobi–Bellman equation. Applications are given to the optimal ergodic control of stochastic partial differential equations.