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arXiv · 2401.09702

The multiplicative ergodic theorem for McKean-Vlasov SDEs

Xianjin Cheng·Zhenxin Liu·Lixin Zhang
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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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Xianjin Cheng, Zhenxin Liu, Lixin Zhang. 2024-01-18. The multiplicative ergodic theorem for McKean-Vlasov SDEs. https://arxiv.org/abs/2401.09702

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