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Ergodicity of Lévy flows

  • Mathematics


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.

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