Stationary distributions of McKean-Vlasov SDEs with jumps: existence, uniqueness, and multiplicity

2025/04/22 by Jianhai Bao, Jian Wang, Bao, Jianhai +1 · 2 citations
Economics, Econometrics and Finance · #FOS: Mathematics #Market Dynamics and Volatility #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2504.15898

openalex publication_date 2025/04/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we are interested in the issues on existence, uniqueness, and multiplicity of stationary distributions for McKean-Vlasov SDEs with jumps. In detail, with regarding to McKean-Vlasov SDEs driven by pure jump Lévy processes, we principally (i) explore the existence of stationary distributions via Schauder's fixed point theorem under an appropriate Lyapunov condition; (ii) tackle the uniqueness of stationary distributions and the convergence to the equilibria as long as the underlying drifts are continuous with respect to the measure variables under the weighted total variation distance and the L¹-Wasserstein distance, respectively; (iii) demonstrate the multiplicity of stationary distributions under a locally dissipative condition. In addition, some illustrative examples are provided to show that the associated McKean-Vlasov SDEs possess a unique, two and three stationary distributions, respectively.

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• A Note on the instability of equilibria for distribution dependent SDEs
• On the infinite time horizon approximation for Lévy-driven McKean-Vlasov SDEs with common noise

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