HJB Equations in Infinite Dimension and Optimal Control of Stochastic Evolution Equations Via Generalized Fukushima Decomposition

A stochastic optimal control problem driven by an abstract evolution equation in a separable Hilbert space is considered. Thanks to the identification of the mild solution of the state equation as a $\nu$-weak Dirichlet process, the value process is proved to be a real weak Dirichlet process. The uniqueness of the corresponding decomposition is used to prove a verification theorem. Through that technique several of the required assumptions are milder than those employed in previous contributions about nonregular solutions of Hamilton--Jacobi--Bellman equations.