[Paper Review] Exponential ergodicity and regularity for equations with Lévy noise

Enrico Priola, Armen Shirikyan+2 more|arXiv (Cornell University)|Feb 27, 2011

Stochastic processes and financial applications28 references4 citations

TL;DR

This paper establishes exponential ergodicity in total variation for stochastic differential equations with α-stable Lévy noise in both finite and infinite-dimensional Hilbert spaces. Using Harris' theorem and Doeblin’s coupling method, it proves that solutions converge exponentially fast to an invariant measure under mild conditions: Hölder continuous drift (with exponent > 1 − α/2) and non-degenerate symmetric α-stable noise (1 < α < 2), without requiring smallness assumptions on the drift.

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. 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 [Priola 2010], with Hölder continuous drift and a general, non-degenerate, symmetric $α$-stable noise, and infinite-dimensional parabolic systems, introduced in [Priola-Zabczyk 2009], with Lipschitz drift and cylindrical $α$-stable noise. We show that if the nonlinearity is bounded, then the processes are exponential mixing. %under the total variation norm. This improves, in particular, an earlier result established in [Priola-Xu-Zabczyk 2010] using the weak convergence induced by the Kantorovich-Wasserstein metric.

Motivation & Objective

• To establish exponential convergence to an invariant measure in total variation norm for SDEs driven by α-stable Lévy noise.
• To extend existing results by removing smallness assumptions on the drift, which were required in prior works using Wasserstein metrics.
• To demonstrate that irreducibility and uniform strong Feller property are sufficient for exponential mixing under mild regularity conditions on drift and noise.
• To provide two distinct proofs—via Harris’ theorem and coupling—highlighting the role of Lyapunov functions and hitting time estimates.
• To generalize earlier results on finite-dimensional SDEs and infinite-dimensional SPDEs with cylindrical α-stable noise.

Proposed method

• Applies Harris' theorem by verifying Lyapunov function conditions and uniform smallness of transition kernels on small sets.
• Employs Doeblin’s coupling argument using a maximal coupling of two solutions starting from different initial conditions.
• Establishes irreducibility and uniform strong Feller property via gradient estimates and support properties of Lévy processes.
• Uses exponential moment bounds on hitting times of small balls to control coupling time distribution.
• Constructs a coupling chain via iterated stopping times τₙ, where solutions are reinitialized upon entering a small ball.
• Applies Chebyshev’s inequality to bound the probability that coupling has not occurred by time kT, leading to exponential decay.

Experimental results

Research questions

• RQ1 Can exponential ergodicity in total variation be established for SDEs with α-stable Lévy noise without smallness assumptions on the drift?
• RQ2 To what extent do irreducibility and uniform strong Feller property ensure exponential mixing in the presence of non-Gaussian Lévy noise?
• RQ3 How do the Harris and coupling approaches compare in proving exponential ergodicity for non-Markovian or non-diffusive SDEs?
• RQ4 What regularity conditions on the drift and noise are necessary and sufficient for exponential mixing in finite and infinite dimensions?
• RQ5 Can hitting time estimates for α-stable processes be used to derive quantitative convergence rates in total variation?

Key findings

• Exponential ergodicity in total variation holds for finite-dimensional SDEs with α-stable noise (1 < α < 2) and Hölder continuous drift with exponent η > 1 − α/2.
• For infinite-dimensional parabolic SPDEs with cylindrical α-stable noise and Lipschitz drift, exponential mixing is established when the nonlinearity is bounded.
• The coupling method yields explicit exponential bounds: P(ρ > kT) ≤ C e^{-ηkT}(1 + |x₁|ᵖ + |x₂|ᵖ) for some η > 0, where ρ is the coupling time.
• The Harris approach relies on verifying Lyapunov and small set conditions, with the uniform strong Feller property ensuring regularity of transition kernels.
• Irreducibility is proven via support properties of α-stable processes and the non-degeneracy of the noise, even in infinite dimensions.
• The results improve upon prior work [30] by replacing weak convergence in Kantorovich–Wasserstein metric with stronger total variation convergence, without requiring smallness of the drift.