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 .In this paper, we study ergodic properties of a Markov process X in R m , given by the SDEc(X (t−), u)ν(dt, du). (0.1)
We construct intrinsic on-and off-diagonal upper and lower estimates for the transition probability density of a Lévy process in small time. By intrinsic we mean that such estimates reflect the structure of the characteristic exponent of the process. The technique used in the paper relies on the asymptotic analysis of the inverse Fourier transform of the respective characteristic function. We provide several examples, in particular, with rather irregular Lévy measure, to illustrate our results.
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