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

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  • Cerrai, Sandra

Abstract

We consider a reaction-diffusion equation in a bounded domain , 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 with values in into the space of differentiable functions from H into . 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.

Suggested Citation

  • Cerrai, Sandra, 1999. "Differentiability of Markov semigroups for stochastic reaction-diffusion equations and applications to control," Stochastic Processes and their Applications, Elsevier, vol. 83(1), pages 15-37, September.
    • Handle: RePEc:eee:spapps:v:83:y:1999:i:1:p:15-37
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    Cited by:

    1. Giorgio Fabbri & Fausto Gozzi & Andrzej Swiech, 2017. "Stochastic Optimal Control in Infinite Dimensions - Dynamic Programming and HJB Equations," Post-Print hal-01505767, HAL.
    2. Albeverio, Sergio & Mastrogiacomo, Elisa & Smii, Boubaker, 2013. "Small noise asymptotic expansions for stochastic PDE’s driven by dissipative nonlinearity and Lévy noise," Stochastic Processes and their Applications, Elsevier, vol. 123(6), pages 2084-2109.

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