Ergodic Control of Semilinear Stochastic Equations and Hamilton-jacobi Equations

In this paper we consider optimal control of stochastic semilinear equations with linearly increasing drift and cylindrical noise. We show existence and uniqueness (up to an additive constant) of solutions to the stationary Hamilton-Jacobi equation associated with the cost functional given by the asymptotic average per unit time cost. As a consequence we nd the optimizing controls given in the feedback form. To obtain these results we prove also some new results on the transition semigroups of semilinear diiusions acting in the spaces of continuous functions with the weighted sup norms and on the optimal control of semilinear diiusions for the discounted cost functional.