  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 ergodic control of stochastic partial differential
equations.
