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.
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Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday
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Gozzi, F. Smoothing properties of nonlinear transition semigroups: case of lipschitz nonlinearities. J. evol. equ. 6, 711–743 (2006). https://doi.org/10.1007/s00028-006-0285-4
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DOI: https://doi.org/10.1007/s00028-006-0285-4