Abstract
We study exponential ergodicity of a broad class of stochastic processes whose dynamics are governed by pure jump Lévy noise. In the first part of the paper we focus on solutions of stochastic differential equations (SDEs) whose drifts satisfy general Lyapunov-type conditions. By applying techniques that combine couplings, appropriately constructed -Wasserstein distances and Lyapunov functions, we show exponential convergence of solutions of such SDEs to their stationary distributions, both in the total variation and Wasserstein distances. The second part of the paper is devoted to SDEs of McKean–Vlasov type with distribution dependent drifts. We prove a uniform in time propagation of chaos result, providing quantitative bounds on convergence rate of interacting particle systems with Lévy noise to the corresponding McKean–Vlasov SDE. Then, extending our techniques from the first part of the paper, we obtain results on exponential ergodicity of solutions of McKean–Vlasov SDEs, under general conditions on the drift and the driving noise.
Nous étudions l’ergodicité exponentielle d’une large classe de processus stochastiques dont la dynamique est régie par un bruit de Lévy à sauts purs. Dans la première partie de l’article, nous nous concentrons sur les solutions d’équations différentielles stochastiques (SDEs) dont les dérives satisfont aux conditions générales de type Lyapunov. En appliquant des techniques qui combinent des couplages, des distances de -Wasserstein et des fonctions de Lyapunov correctement construites, nous montrons une convergence exponentielle des solutions de telles SDE vers leurs distributions stationnaires, à la fois sous la distance de variation totale et de Wasserstein. La deuxième partie de l’article est consacrée aux SDEs de type McKean–Vlasov avec des dérives dépendant de la distribution. Nous prouvons un résultat de propagation de chaos uniformement en temps, en fournissant des bornes quantitatives sur le taux de convergence des systèmes de particules en interaction avec le bruit de Lévy vers le SDE deMcKean–Vlasov correspondant. Ensuite, en étendant nos techniques dans la première partie de l’article, nous obtenons des résultats sur l’ergodicité exponentielle des solutions de SDE de McKean–Vlasov, dans des conditions générales sur la dérive et le bruit.
Acknowledgements
We thank the two anonymous referees for their helpful comments and valuable corrections that have led to significant improvements of the presentation in the article. The research of Mingjie Liang is supported by the Science Foundation of the Education Department of Fujian Province (No. JAT190701) and the National Science Foundation of Sanming University (No. B201914). The majority of this work was completed when Mateusz B. Majka was affiliated to the University of Warwick and supported by the EPSRC grant no. EP/P003818/1. The research of Jian Wang is supported by the National Natural Science Foundation of China (Nos. 11831014 and 12071076), the Program for Probability and Statistics: Theory and Application (No. IRTL1704) and the Program for Innovative Research Team in Science and Technology in Fujian Province University (IRTSTFJ).
Citation
Mingjie Liang. Mateusz B. Majka. Jian Wang. "Exponential ergodicity for SDEs and McKean–Vlasov processes with Lévy noise." Ann. Inst. H. Poincaré Probab. Statist. 57 (3) 1665 - 1701, August 2021. https://doi.org/10.1214/20-AIHP1123
Information