Infinite horizon stochastic optimal control problems with degenerate noise and elliptic equations in Hilbert spaces (Q996064)

Solutions of a semilinear elliptic equation involving the generator of a Hilbert space valued Markov process are established under conditions on the directional derivatives of the associated transition semigroup. This is applied to the Hamilton-Jacobi-Bellman equation for a controlled stochastic evolution equation with infinite horizon discounted cost. Controlled stochastic nonlinear heat equation and wave equation are treated as special cases.