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L 2-theory for non-symmetric Ornstein–Uhlenbeck semigroups on domains

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Abstract

We prove that the mild solution of the stochastic evolution equation \({{d}X(t) = AX(t)\,{d}t + {d}W(t)}\) on a Banach space E has a continuous modification if the associated Ornstein–Uhlenbeck semigroup is analytic on L 2 with respect to the invariant measure. This result is used to extend recent work of Da Prato and Lunardi for Ornstein–Uhlenbeck semigroups on domains \({\mathcal{O} \subseteq E}\) to the non-symmetric case. Denoting the generator of the Ornstein–Uhlenbeck semigroup by \({L_\mathcal{O}}\), we obtain sufficient conditions in order that the domain of \({\sqrt{-L_\mathcal{O}}}\) be a first-order Sobolev space.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Faculty of Natural and Applied Sciences, Notre Dame University-Louaize, P.O. Box 72, Zouk Mikael, Zouk Mosbeh, Lebanon

    Joyce Assaad

  2. Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA, Delft, The Netherlands

    Jan van Neerven

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  1. Joyce Assaad
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  2. Jan van Neerven
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Correspondence to Jan van Neerven.

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The second named author is supported by VICI subsidy 639.033.604 of the Netherlands Organisation for Scientific Research (NWO).

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Assaad, J., van Neerven, J. L 2-theory for non-symmetric Ornstein–Uhlenbeck semigroups on domains. J. Evol. Equ. 13, 107–134 (2013). https://doi.org/10.1007/s00028-012-0171-1

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