Smoothing properties of nonlinear transition semigroups: case of lipschitz nonlinearities

Abstract.

We consider a semilinear stochastic differential equation in a Hilbert space H with a Lipschitz continuous (possibly unbounded) nonlinearity F. We prove that the associated transition semigroup {P _t , t ≥  0}, acting on the space of bounded measurable functions from H to $\mathbb{R}$, transforms bounded nondifferentiable functions into Fréchet differentiable ones. Moreover we consider the associated Kolmogorov equation and we prove that it possesses a unique “strong” solution (where “strong” means limit of classical solutions) given by the semigroup {P _t , t ≥  0} applied to the initial condition. This result is a starting point to prove existence and uniqueness of strong solutions to Hamilton - Jacobi - Bellman equations arising in control theory.