Differentiability of Markov semigroups for stochastic reaction–diffusion equations and applications to control

We consider a reaction–diffusion equation in a bounded domain O⊂Rd, driven by a space–time white noise, with a drift term having polynomial growth and a diffusion term which is not boundedly invertible, in general. We are showing that the transition semigroup corresponding to the equation has a regularizing effect. More precisely, we show that it maps bounded and Borel functions defined in the Hilbert space H=L2(O) with values in R into the space of differentiable functions from H into R. An estimate for the sup-norm of the derivative of the semigroup is given. We apply these results to the study of the corresponding Hamilton–Jacobi equation arising in stochastic control theory.