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We solve in mild sense Hamilton Jacobi Bellman equations, both in an infinite dimensional Hilbert space and in a Banach space, with Lipschitz Hamiltonian and Lipschitz continuous final condition, and asking only a weak regularizing property on the transition semigroup of the corresponding state equation. The results are applied to solve stochastic optimal control problems; the models we can treat include a controlled stochastic heat equation in space dimension one and with control and noise on a subdomain.

HJB equations in infinite dimensions under weak regularizing properties