Exponential ergodicity of the solutions to SDE’s with a jump noise

Mild sufficient conditions for exponential ergodicity of a Markov process defined as the solution to a SDE with jump noise are given. These conditions include three principal claims: recurrence condition R , topological irreducibility condition S and non-degeneracy condition N , the latter formulated in terms of a certain random subspace of R m , associated with the initial equation. Examples are given, showing that, in general, none of the principal claims can be removed without losing ergodicity of the process. The key point in the approach developed in the paper is that the local Doeblin condition can be derived from N and S via the stratification method and a criterium for the convergence in variation of the family of induced measures on R m .