Global Regular Solutions of Second Order Hamilton–Jacobi Equations in Hilbert Spaces with Locally Lipschitz Nonlinearities

Abstract This paper is devoted to the study of a second order Hamilton–Jacobi equation in infinite dimensions with a locally Lipschitz continuous Hamiltonian H , related with a stochastic optimal control problem driven by white noise. In an earlier paper the author studied the case of globally Lipschitz continuous Hamiltonian H . This paper is concerned with the case of locally Lipschitz continuous H that include e.g., the quadratic case, which appears very frequently in the applications. Existence, uniqueness, and C 2 regularity of a local solution follow by methods given previously. The heart of the work is the proof of a priori estimates of the C 1 -norm of the local solution. This has been done by studying properties of nonlinear transition semigroups and by a careful application of them to our problem. The results have been applied to a stochastic optimal control problem proving existence of feedback optimal controls.