Ergodicity of Lévy flows

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 (ζj, 1⩽j⩽n) of complete smooth vector fields on M and an n-dimensional Lévy process X with characteristics (b,σ,ν), where b=(bj) is a real vector, σ=(σij) is a real matrix, 1⩽j⩽n, 1⩽i⩽m, m⩽n and ν is a Lévy measure on Rn−{0}. The induced flows of local diffeomorphisms (γt(.,w), t⩾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.