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Generalized Mehler semigroups and applications
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  • Published: June 1996

Generalized Mehler semigroups and applications

  • Vladimir I. Bogachev1,
  • Michael Röckner2 &
  • Byron Schmuland3 

Probability Theory and Related Fields volume 105, pages 193–225 (1996)Cite this article

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Summary

We construct and study generalized Mehler semigroups (p t ) t ≧0 and their associated Markov processesM. The construction methods for (p t ) t ≧0 are based on some new purely functional analytic results implying, in particular, that any strongly continuous semigroup on a Hilbert spaceH can be extended to some larger Hilbert spaceE, with the embeddingH⊂E being Hilbert-Schmidt. The same analytic extension results are applied to construct strong solutions to stochastic differential equations of typedX t =C dW t +AX t dt (with possibly unbounded linear operatorsA andC onH) on a suitably chosen larger spaceE. For Gaussian generalized Mehler semigroups (p t ) t ≧0 with corresponding Markov processM, the associated (non-symmetric) Dirichlet forms (E D(E)) are explicitly calculated and a necessary and sufficient condition for path regularity ofM in terms of (E,D(E)) is proved. Then, using Dirichlet form methods it is shown thatM weakly solves the above stochastic differential equation if the state spaceE is chosen appropriately. Finally, we discuss the differences between these two methods yielding strong resp. weak solutions.

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Authors and Affiliations

  1. Department of Mechanics and Mathematics, Moscow State University, 119899, Moscow, Russia

    Vladimir I. Bogachev

  2. Department of Mathematics, University of Bielefeld, D-33615, Bielefeld, Germany

    Michael Röckner

  3. Department of Mathematical Sciences, University of Alberta, T6G 2G1, Edmonton, Alberta, Canada

    Byron Schmuland

Authors
  1. Vladimir I. Bogachev
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  2. Michael Röckner
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  3. Byron Schmuland
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Bogachev, V.I., Röckner, M. & Schmuland, B. Generalized Mehler semigroups and applications. Probab. Th. Rel. Fields 105, 193–225 (1996). https://doi.org/10.1007/BF01203835

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  • Received: 05 October 1995

  • Revised: 12 December 1995

  • Issue Date: June 1996

  • DOI: https://doi.org/10.1007/BF01203835

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Mathematics Subject Classification (1991)

  • 47D07
  • 60H15
  • 31C25
  • 60H10
  • 60G15
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