  This paper is about operator-theoretic methods for solving nonlinear
stochastic optimal control problems to global optimality. These methods
leverage on the convex duality between optimally controlled diffusion processes
and Hamilton-Jacobi-Bellman (HJB) equations for nonlinear systems in an ergodic
Hilbert-Sobolev space. In detail, a generalized Bakry-Emery condition is
introduced under which one can establish the global exponential stabilizability
of a large class of nonlinear systems. It is shown that this condition is
sufficient to ensure the existence of solutions of the ergodic HJB for
stochastic optimal control problems on infinite time horizons. Moreover, a
novel dynamic programming recursion for bounded linear operators is introduced,
which can be used to numerically solve HJB equations by a Galerkin projection.
