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Transition Semigroups of Banach Space-Valued Ornstein–Uhlenbeck Processes

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Abstract

We investigate the transition semigroup of the solution to a stochastic evolution equation dX(t)=AX(t) dt+dW H (t), t≥0, where A is the generator of a C 0-semigroup S on a separable real Banach space E and W H (t) t≥0 is cylindrical white noise with values in a real Hilbert space H which is continuously embedded in E. Various properties of these semigroups, such as the strong Feller property, the spectral gap property, and analyticity, are characterized in terms of the behaviour of S in H. In particular we investigate the interplay between analyticity of the transition semigroup, S-invariance of H, and analyticity of the restricted semigroup S H .

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Authors and Affiliations

  1. School of Mathematics, The University of New South Wales, Sydney, 2052, Australia

    B. Goldys

  2. Department of Applied Mathematical Analysis, Technical University of Delft, P.O. Box 5031, 2600, GA Delft, The Netherlands

    J. M. A. M. van Neerven

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  1. B. Goldys
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  2. J. M. A. M. van Neerven
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Goldys, B., van Neerven, J.M.A.M. Transition Semigroups of Banach Space-Valued Ornstein–Uhlenbeck Processes. Acta Applicandae Mathematicae 76, 283–330 (2003). https://doi.org/10.1023/A:1023261101091

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