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Markov Processes with Jumps on Manifolds and Lie Groups

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Geometry and Invariance in Stochastic Dynamics (RTISD19 2019)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 378))

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Abstract

We review some developments concerning Markov and Feller processes with jumps in geometric settings. These include stochastic differential equations in Markus canonical form, the Courrège theorem on Lie groups, and invariant Markov processes on manifolds under both transitive and more general Lie group actions.

We dedicate this article to the memory of Hiroshi Kunita (1937–2019).

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Notes

  1. 1.

    Our assumptions on \((E, {\mathcal E})\) ensure that such a version exists.

  2. 2.

    In fact he obtained the representation on a larger domain, but we will not need that here.

  3. 3.

    We say that \(\mu \) is a Lévy kernel if \(\mu (\sigma , \cdot )\) is a Lévy measure for all \(\sigma \in G\).

  4. 4.

    K is independent of the choice of point chosen, up to isomorphism.

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Acknowledgements

We thank the referee for some helpful suggestions.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, University of Sheffield, Sheffield, S3 7RH, UK

    David Applebaum

  2. Department of Mathematics, Auburn University, Auburn, AL, USA

    Ming Liao

Authors
  1. David Applebaum
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  2. Ming Liao
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Corresponding author

Correspondence to David Applebaum .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Università degli Studi di Milano, Milan, Italy

    Stefania Ugolini

  2. Department of Mathematics, Università degli Studi di Milano, Milan, Italy

    Marco Fuhrman

  3. Department of Economics, Università degli Studi dell'Insubria, Varese, Italy

    Elisa Mastrogiacomo

  4. Department of Agricultural and Environmental Sciences, Università degli Studi di Milano, Milan, Italy

    Paola Morando

  5. Mathematic and Informatic Department, Bergische Universität Wuppertal, Wuppertal, Germany

    Barbara Rüdiger

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Applebaum, D., Liao, M. (2021). Markov Processes with Jumps on Manifolds and Lie Groups. In: Ugolini, S., Fuhrman, M., Mastrogiacomo, E., Morando, P., Rüdiger, B. (eds) Geometry and Invariance in Stochastic Dynamics. RTISD19 2019. Springer Proceedings in Mathematics & Statistics, vol 378. Springer, Cham. https://doi.org/10.1007/978-3-030-87432-2_2

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