HJB equations in infinite dimension and optimal control of stochastic evolution equations via eneralized 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 V-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.