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Ergodicity of homogeneous Brownian flows

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  • Mohari, Anilesh

Abstract

Let M be a finite-dimensional smooth-oriented paracompact manifold and [zeta]k, 0[less-than-or-equals, slant]k[less-than-or-equals, slant]d, be a family of complete smooth vector fields on M so that the Brownian flow associated with exists globally. We prove that any volume form [mu] on M is irreducible for the Brownian flows if and only if there exists only constant functions [psi][set membership, variant]L[infinity](M,[mu]) satisfying the following equation:[psi]=[psi]o[alpha]([zeta]k,t) [for all]t[set membership, variant]R, 0[less-than-or-equals, slant]k[less-than-or-equals, slant]d,where ([alpha]([zeta],t) [for all]t[set membership, variant]R) is the one-parameter group of diffeomorphism on M associated with the complete vector field [zeta]. In such a case, an invariant finite volume form [mu] is ergodic for the flow.

Suggested Citation

  • Mohari, Anilesh, 2003. "Ergodicity of homogeneous Brownian flows," Stochastic Processes and their Applications, Elsevier, vol. 105(1), pages 99-116, May.
    • Handle: RePEc:eee:spapps:v:105:y:2003:i:1:p:99-116
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    Cited by:

    1. Mohari, Anilesh, 2004. "Ergodicity of Lévy flows," Stochastic Processes and their Applications, Elsevier, vol. 112(2), pages 245-259, August.
    2. 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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