Infinite dimensional weak Dirichlet processes, stochastic PDEs and optimal control

The present paper continues the study of infinite dimensional calcu-lus via regularization, started by C. Di Girolami and the second named author, introducing the notion of weak Dirichlet process in this context. Such a process X, taking values in a Hilbert space H, is the sum of a local martingale and a suitable orthogonal process. The new concept is shown to be useful in several contexts and directions. On one side, the mentioned decomposition appears to be a substitute of an Itô’s type formula applied to f(t,X(t)) where f: [0, T] ×H → R is a C0,1 function and, on the other side, the idea of weak Dirichlet process fits the widely used notion of mild solution for stochastic PDE. As a specific application, we provide a verification theorem for stochastic optimal con-trol problems whose state equation is an infinite dimensional stochastic evolution equation