Di erentiability of Markov semigroups for stochastic reaction–di usion equations and applications to control

We consider a reaction–di usion equation in a bounded domain O⊂Rd, driven by a space– time white noise, with a drift term having polynomial growth and a di usion term which is not boundedly invertible, in general. We are showing that the transition semigroup corresponding to the equation has a regularizing e ect. More precisely, we show that it maps bounded and Borel functions de ned in the Hilbert space H=L(O) with values in R into the space of di erentiable 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. c © 1999 Elsevier Science B.V. All rights reserved. MSC: 60H15; 60J25; 93E20; 93C20