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.

Differentiability of Markov semigroup for stochastic reaction-diffusion equations and applications to control / S. CERRAI. - In: STOCHASTIC PROCESSES AND THEIR APPLICATIONS. - ISSN 0304-4149. - STAMPA. - 83:(1999), pp. 15-37. [10.1016/S0304-4149(99)00014-9]

Differentiability of Markov semigroup for stochastic reaction-diffusion equations and applications to control

CERRAI, SANDRA
1999

Abstract

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.
1999
83
15
37
S. CERRAI
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Utilizza questo identificatore per citare o creare un link a questa risorsa: https://hdl.handle.net/2158/312624
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