Exponential ergodicity and regularity for equations with Lévy noise

Abstract

We prove exponential convergence to the invariant measure, in the total variation norm, for solutions of SDEs driven by α-stable noises in finite and in infinite dimensions. Two approaches are used. The first one is based on Harris theorem, and the second on Doeblin's coupling argument [10]. Irreducibility, Lyapunov function techniques, and uniform strong Feller property play an essential role in both approaches. We concentrate on two classes of Markov processes: solutions of finite-dimensional equations, introduced in [29], with Hölder continuous drift and a general, non-degenerate, symmetric α-stable noise, and infinite-dimensional parabolic systems, introduced in [32], with Lipschitz drift and cylindrical α-stable noise. We show that if the nonlinearity is bounded, then the processes are exponential mixing. This improves, in particular, an earlier result established in [30] using the weak convergence induced by the Kantorovich-Wasserstein metric.

Irreducibility and Exponential Mixing of Some Stochastic Hydrodynamical Systems Driven by Pure Jump Noise

“…Now we derive that for any real number R > 0, t > 0 there exists a constant K such that sup n∈N, x∈H n |x|≤1 [48]. Let δ the constant given in Proposition 5.1.…”

“…To prove our results we were inspired by [46,48] and [49], which treated stable processes driven SPDEs with bounded nonlinearity which does not include the examples we treat in this paper. It is clear from the sketch of our results and the assumptions on our driving noise that irreducibility and ergodicity for 2D Navier-Stokes, Magnetohydrodynamics equations and the 3-dimensional Leray-α driven by stable processes do not follow from our results and are still an open problem.…”

“…The noise we consider in this paper is non-degenerate, i.e., β k > 0 for all k ∈ N and it was pointed out by one of the anonymous referees that it might also be possible to establish Theorem 3.7 by using the Harris approach (see, for instance, [48]) which would considerably simplify and shorten the proof.…”

“…However, several results about the qualitative or long-time behavior of solution of SPDEs driven by Lévy noise have been obtained during the last decade. We refer, among others, to [18,19,33,38,43,45,46,48,49] for results related to the ergodicity, irreducibility and mixing property of several classes of stochastic evolution equations driven by Lévy processes.…”

Ergodicity for the 3D stochastic Navier–Stokes equations perturbed by Lévy noise

“…Since Q is nondegenerate and the jump noise coefficient is independent of the state, ( P ) ≥0 has a unique invariant measure (see [1,37]). By Dynkin's formula, we have…”