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Ergodicity of Lévy flows

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  • Mohari, Anilesh

Abstract

We consider a stochastic differential equation (SDE) of jump type on a finite-dimensional connected smooth and oriented manifold M. The SDE is driven by a family ([zeta]j, 1[less-than-or-equals, slant]j[less-than-or-equals, slant]n) of complete smooth vector fields on M and an n-dimensional Lévy process X with characteristics (b,[sigma],[nu]), where b=(bj) is a real vector, [sigma]=([sigma]ij) is a real matrix, 1[less-than-or-equals, slant]j[less-than-or-equals, slant]n, 1[less-than-or-equals, slant]i[less-than-or-equals, slant]m, m[less-than-or-equals, slant]n and [nu] is a Lévy measure on . The induced flows of local diffeomorphisms ([gamma]t(.,w), t[greater-or-equal, slanted]0) on M are assumed to be stochastically complete. We find a necessary and sufficient condition for irreducibility of the flows with respect to a volume measure. We apply this criterion to the Horizontal Lévy flows on the orthonormal frame bundle over a compact Riemannian manifold and prove that the spherical symmetric (isotropic) Lévy motion on M is ergodic with respect to the Riemannian measure on M.

Suggested Citation

  • Mohari, Anilesh, 2004. "Ergodicity of Lévy flows," Stochastic Processes and their Applications, Elsevier, vol. 112(2), pages 245-259, August.
    • Handle: RePEc:eee:spapps:v:112:y:2004:i:2:p:245-259
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    References listed on IDEAS

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    1. Mohari, Anilesh, 2003. "Ergodicity of homogeneous Brownian flows," Stochastic Processes and their Applications, Elsevier, vol. 105(1), pages 99-116, May.
    2. Applebaum, David & Tang, Fuchang, 2001. "Stochastic flows of diffeomorphisms on manifolds driven by infinite-dimensional semimartingales with jumps," Stochastic Processes and their Applications, Elsevier, vol. 92(2), pages 219-236, April.
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    Cited by:

    1. David Applebaum & Rosemary Shewell Brockway, 2021. "$$L^2$$ L 2 Properties of Lévy Generators on Compact Riemannian Manifolds," Journal of Theoretical Probability, Springer, vol. 34(2), pages 1029-1042, June.

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    1. David Applebaum & Rosemary Shewell Brockway, 2021. "$$L^2$$ L 2 Properties of Lévy Generators on Compact Riemannian Manifolds," Journal of Theoretical Probability, Springer, vol. 34(2), pages 1029-1042, June.

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