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

Jianhai Bao, Jian Wang

TL;DR

The work addresses existence, uniqueness, and multiplicity of stationary distributions for McKean–Vlasov SDEs with jumps driven by pure-jump Lévy noise. It develops a fixed-point framework based on frozen SDEs and a Lyapunov control to prove existence, and introduces a locally dissipative condition to generate multiple equilibria, complemented by gradient-estimate techniques to obtain contraction and exponential ergodicity in total variation. The main results show existence under $(\mathbf{A}_1)$–$(\mathbf{A}_3)$, multiplicity under an additional local dissipation $(\mathbf{A}_4)$, and uniqueness with exponential convergence under $(\mathbf{A}_5)$–$(\mathbf{A}_6)$, with explicit one- and multi-dimensional examples illustrating phase transitions between 1, 2, and 3 stationary distributions. The methods extend prior Brownian-motion results to jump-driven systems, providing verifiable criteria, moment bounds, and detailed constructions that illuminate the behavior of distribution-dependent SDEs with jumps in both existence and ergodicity regimes.

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 $L^1$-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.

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

TL;DR

The work addresses existence, uniqueness, and multiplicity of stationary distributions for McKean–Vlasov SDEs with jumps driven by pure-jump Lévy noise. It develops a fixed-point framework based on frozen SDEs and a Lyapunov control to prove existence, and introduces a locally dissipative condition to generate multiple equilibria, complemented by gradient-estimate techniques to obtain contraction and exponential ergodicity in total variation. The main results show existence under –, multiplicity under an additional local dissipation , and uniqueness with exponential convergence under –, with explicit one- and multi-dimensional examples illustrating phase transitions between 1, 2, and 3 stationary distributions. The methods extend prior Brownian-motion results to jump-driven systems, providing verifiable criteria, moment bounds, and detailed constructions that illuminate the behavior of distribution-dependent SDEs with jumps in both existence and ergodicity regimes.

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 -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.

Paper Structure

This paper contains 15 sections, 9 theorems, 199 equations.

Table of Contents

  1. Background and main results
  2. Background
  3. Main results
  4. Existence of stationary distributions
  5. Multiplicity of stationary distributions
  6. Uniqueness of stationary distributions
  7. Proof of Theorem \ref{['thm1']} and verification of Assumption (${\bf A}_3$)
  8. Qualitative analysis of $\pi^\mu$
  9. Proof of Theorem \ref{['thm1']}
  10. Verification of Assumption $({\bf A}_3)$
  11. Proofs of Theorem \ref{['thm3']}, Example \ref{['exm']} and example \ref{['exm2']}
  12. proofs of theorem \ref{['thm4']}, Corollary \ref{['cor']} and Example \ref{['exa3']}
  13. Appendix
  14. Uniqueness of stationary distributions via the $L^1$-Wasserstein distance
  15. Verification of \ref{['UT']} in Assumption (${\bf A}_{5}$)

Key Result

Theorem 1.1

Under Assumptions $({\bf A}_1)$, $({\bf A}_2)$ and $({\bf A}_3)$, E1 has a stationary distribution $\pi$ with finite $\beta^*$-th moment $($i.e., $\pi(|\cdot|^{\beta^*})<\infty$$)$.

Theorems & Definitions (29)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Example 1.5
  • Example 1.6
  • Theorem 1.7
  • Remark 1.8
  • Corollary 1.9
  • Example 1.10
  • ...and 19 more