Title: HJB equations in infinite dimension and optimal control of stochastic evolution equations via generalized Fukushima decomposition,
Abstract: 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 $\nu$-weak Dirichlet process, the value
processes 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 non-regular solutions of
Hamilton-Jacobi-Bellman equations.