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The multiplicative ergodic theorem for McKean-Vlasov SDEs

Xianjin Cheng, Zhenxin Liu, Lixin Zhang

TL;DR

The paper extends multiplicative ergodic theory to McKean–Vlasov SDEs by defining Lyapunov exponents via $\limsup_{t\to\infty}\frac{1}{t}\log|\partial_x\Phi_{0,t}(\omega,x)v|$, addressing the absence of a flow in mean-field dynamics. It develops a decoupled, two-component framework that separates deterministic spatial evolution from distributional effects, and establishes a mean-field MET: a finite Lyapunov spectrum on a full-measure set with exponents bounded by a constant $\kappa$. The analysis leverages Cantelli's SLLN and Hájek–Rényi inequalities to control stochastic growth and shows that a uniform, trajectory-wise approach is essential due to lack of flow properties. An explicit example demonstrates that the limit may not exist even for regular coefficients, underscoring the critical role of distribution dependence in MV-SDEs and justifying the limsup-based definition for Lyapunov exponents in the mean-field MET.

Abstract

In this paper, we establish the multiplicative ergodic theorem for McKean-Vlasov stochastic differential equations, in which the Lyapunov exponent is defined using the upper limit. The reasonability of this definition is illustrated through an example; i.e., even when the coefficients are regular enough and their first-order derivatives are bounded, the upper limit cannot be replaced by a limit, as the limit may not exist. Furthermore, the example reveals how the dependence on distribution significantly influences the dynamics of the system and evidently distinguishes McKean-Vlasov stochastic differential equations from classical stochastic differential equations.

The multiplicative ergodic theorem for McKean-Vlasov SDEs

TL;DR

The paper extends multiplicative ergodic theory to McKean–Vlasov SDEs by defining Lyapunov exponents via , addressing the absence of a flow in mean-field dynamics. It develops a decoupled, two-component framework that separates deterministic spatial evolution from distributional effects, and establishes a mean-field MET: a finite Lyapunov spectrum on a full-measure set with exponents bounded by a constant . The analysis leverages Cantelli's SLLN and Hájek–Rényi inequalities to control stochastic growth and shows that a uniform, trajectory-wise approach is essential due to lack of flow properties. An explicit example demonstrates that the limit may not exist even for regular coefficients, underscoring the critical role of distribution dependence in MV-SDEs and justifying the limsup-based definition for Lyapunov exponents in the mean-field MET.

Abstract

In this paper, we establish the multiplicative ergodic theorem for McKean-Vlasov stochastic differential equations, in which the Lyapunov exponent is defined using the upper limit. The reasonability of this definition is illustrated through an example; i.e., even when the coefficients are regular enough and their first-order derivatives are bounded, the upper limit cannot be replaced by a limit, as the limit may not exist. Furthermore, the example reveals how the dependence on distribution significantly influences the dynamics of the system and evidently distinguishes McKean-Vlasov stochastic differential equations from classical stochastic differential equations.
Paper Structure (9 sections, 8 theorems, 109 equations)

This paper contains 9 sections, 8 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Wiener Measure and Wiener Shift
  4. Lions Derivatives and Regularity Conditions
  5. Cantelli' Strong Law of Large Numbers and the Hájek-Rényi Inequality
  6. The Mean-Field Version of the Multiplicative Ergodic Theorem
  7. Some preparations
  8. Main results
  9. An Example

Key Result

Proposition 2.3

Let $\{\xi_n\}_{n\geq1}$ be a sequence of independent random variables with finite fourth moments and satisfy for some constant $C$. Then as $n\rightarrow+\infty$, where $S_n=\xi_1+...+\xi_n$.

Theorems & Definitions (19)

  • Definition 2.1
  • Proposition 2.3: Cantelli' strong law of large numbers
  • Proposition 2.4: The Hájek-Rényi inequality
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 9 more