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Log-Sobolev Inequalities and Exponential Ergodicity for Non-degenerate and Degenerate McKean-Vlasov SDEs

Xing Huang, Eva Kopfer, Panpan Ren

TL;DR

The article advances the study of McKean-Vlasov SDEs by proving uniform log-Sobolev inequalities for time-inhomogeneous semigroups with frozen distributions and leveraging Wang's Harnack inequality to derive exponential ergodicity in both $L^2$-Wasserstein distance and relative entropy. It develops a framework that accommodates both non-degenerate and degenerate diffusion, and extends to partially dissipative drifts, enabling contraction and entropy decay without requiring explicit invariant measures. The approach hinges on explicit LS bounds tied to the frozen system and uses log-Harnack and Talagrand-type inequalities to transfer these bounds to the evolving McKean-Vlasov dynamics, yielding both existence and uniqueness of invariant measures under small interaction terms and quantitative convergence rates. The paper also provides well-posedness results and two approximation schemes (Yosida and mollifier) to justify the LS framework and facilitate practical analysis of these mean-field systems.

Abstract

The exponential ergodicity of partially dissipative McKean-Vlasov SDEs in the \(L^1\)-Wasserstein distance has been extensively studied using asymptotic reflection coupling. However, the reflection coupling method is not applicable for the exponential ergodicity in $L^2$-Wasserstein distance and relative entropy. In this paper, we first establish uniform log-Sobolev inequalities (in the frozen measure variable with bounded second moments) for the invariant probability measure of the corresponding SDEs with frozen distribution. Second, for the McKean-Vlasov SDEs, we combine the log-Harnack inequality and Talagrand's inequality to derive exponential ergodicity in both $L^2$-Wasserstein distance and relative entropy. Furthermore, we extend these main results to the case of degenerate diffusion.

Log-Sobolev Inequalities and Exponential Ergodicity for Non-degenerate and Degenerate McKean-Vlasov SDEs

TL;DR

The article advances the study of McKean-Vlasov SDEs by proving uniform log-Sobolev inequalities for time-inhomogeneous semigroups with frozen distributions and leveraging Wang's Harnack inequality to derive exponential ergodicity in both -Wasserstein distance and relative entropy. It develops a framework that accommodates both non-degenerate and degenerate diffusion, and extends to partially dissipative drifts, enabling contraction and entropy decay without requiring explicit invariant measures. The approach hinges on explicit LS bounds tied to the frozen system and uses log-Harnack and Talagrand-type inequalities to transfer these bounds to the evolving McKean-Vlasov dynamics, yielding both existence and uniqueness of invariant measures under small interaction terms and quantitative convergence rates. The paper also provides well-posedness results and two approximation schemes (Yosida and mollifier) to justify the LS framework and facilitate practical analysis of these mean-field systems.

Abstract

The exponential ergodicity of partially dissipative McKean-Vlasov SDEs in the -Wasserstein distance has been extensively studied using asymptotic reflection coupling. However, the reflection coupling method is not applicable for the exponential ergodicity in -Wasserstein distance and relative entropy. In this paper, we first establish uniform log-Sobolev inequalities (in the frozen measure variable with bounded second moments) for the invariant probability measure of the corresponding SDEs with frozen distribution. Second, for the McKean-Vlasov SDEs, we combine the log-Harnack inequality and Talagrand's inequality to derive exponential ergodicity in both -Wasserstein distance and relative entropy. Furthermore, we extend these main results to the case of degenerate diffusion.

Paper Structure

This paper contains 27 sections, 15 theorems, 200 equations.

Table of Contents

  1. Introduction
  2. Backgroud on McKean-Vlasov SDEs
  3. Related works
  4. Log-Sobolev inequalities for McKean-Vlasov SDEs
  5. Exponential ergodicity for McKean-Vlasov SDEs
  6. Outline
  7. Non-degenerate case
  8. Log-Sobolev inequality for $P_{s,t}^\mu$ and $P_t^\ast\mu_0$
  9. Time-homogeneous decoupled SDEs
  10. Invariant probability measure
  11. Wang's Harnack inequality and hyperboundedness
  12. Log-Sobolev inequality
  13. Exponential ergodicity of McKean-Vlasov SDEs
  14. Degenerate case
  15. Log-Sobolev inequality for $P_{s,t}^\mu$ and $P_t^\ast\mu_0$
  16. ...and 12 more sections

Key Result

Theorem 2.1

Assume ${\bf (A)}$ and suppose that $\sigma_t(x,\gamma)=\sigma_t(\gamma)$ satisfies for some constant $\delta_1>0$, Then, $P_{s,t}^\mu$ satisfies the log-Sobolev inequality: for any $\mu_\cdot\in C([0,\infty),\mathscr P_2(\mathbb R^d))$ and $f\in C_b^\infty(\mathbb R^d)$, $f>0$, If moreover $\gamma\in\mathscr P_2(\mathbb R^d)$ satisfies the log-Sobolev inequality: for some $L_0\geq 0$, then $P_

Theorems & Definitions (44)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • proof
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7
  • proof
  • Remark 2.8
  • ...and 34 more