Classical Adjoints for Ergodic Stochastic Control

In this paper we consider ergodic optimal control of a diffusion process $\{X_t^u{\}}_{t\ge 0}$, taking values in ${\text{bR}}^n$, where both drift and volatility are controlled. We establish a novel strong duality between the existence of a unique solution to the infinite horizon adjoint BSDE and strong dissipativity of $X^u$. We then proceed to show that the latter implies irreducibility, the strong Feller property and exponential ergodicity. We conclude by discussing the connection with ergodic BSDEs.