Exponential ergodicity for SDEs and McKean-Vlasov processes with Lévy noise

Mingjie Liang; Mateusz B. Majka; Jian Wang

doi:10.1214/20-AIHP1123

Exponential ergodicity for SDEs and McKean-Vlasov processes with Lévy noise

Mingjie Liang, Mateusz B. Majka, Jian Wang

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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 L¹-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.

Original language	English
Pages (from-to)	1665-1701
Number of pages	37
Journal	Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
Volume	57
Issue number	3
Early online date	22 Jul 2021
DOIs	https://doi.org/10.1214/20-AIHP1123
Publication status	Published - Aug 2021

Keywords

Coupling
Exponential ergodicity
Lévy noise
McKean–Vlasov process
Mean-field SDE
Propagation of chaos

ASJC Scopus subject areas

Statistics and Probability
Statistics, Probability and Uncertainty

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10.1214/20-AIHP1123

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Copyright © 2021 Association des Publications de l’Institut Henri Poincaré Ann. Inst. H. Poincaré Probab. Statist. 57 (3) 1665 - 1701, August 2021. https://doi.org/10.1214/20-AIHP1123
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title = "Exponential ergodicity for SDEs and McKean-Vlasov processes with L{\'e}vy noise",

abstract = "We study exponential ergodicity of a broad class of stochastic processes whose dynamics are governed by pure jump L{\'e}vy 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 L1-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 L{\'e}vy 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.",

keywords = "Coupling, Exponential ergodicity, L{\'e}vy noise, McKean–Vlasov process, Mean-field SDE, Propagation of chaos",

author = "Mingjie Liang and Majka, {Mateusz B.} and Jian Wang",

note = "Funding Information: 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). Publisher Copyright: {\textcopyright} Association des Publications de l{\textquoteright}Institut Henri Poincar{\'e}, 2021.",

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AU - Liang, Mingjie

AU - Majka, Mateusz B.

AU - Wang, Jian

N1 - Funding Information: 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). Publisher Copyright: © Association des Publications de l’Institut Henri Poincaré, 2021.

PY - 2021/8

Y1 - 2021/8

N2 - 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 L1-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.

AB - 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 L1-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.

KW - Coupling

KW - Exponential ergodicity

KW - Lévy noise

KW - McKean–Vlasov process

KW - Mean-field SDE

KW - Propagation of chaos

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SN - 0246-0203

VL - 57

SP - 1665

EP - 1701

JO - Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

JF - Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

IS - 3

ER -

Exponential ergodicity for SDEs and McKean-Vlasov processes with Lévy noise

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