The multiplicative ergodic theorem for McKean-Vlasov SDEs

2024/01/18 by Xianjin Cheng, Cheng, Xianjin, Zhenxin Liu +3
Economics, Econometrics and Finance · Physics and Astronomy · #34D08 #37A50 #37H15 #60H10 #Advanced Thermodynamics and Statistical Mechanics #Dynamical Systems (math.DS) #Economic theories and models #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2401.09702

openalex publication_date 2024/01/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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