Long Time Entropy-Cost type Propagation of Chaos

TL;DR

This paper investigates how stochastic noise influences the long-term propagation of chaos in mean field particle systems, demonstrating entropy-based regularization effects and extending results to path-dependent cases without relying on log-Sobolev inequalities.

Contribution

It introduces a novel entropy-cost framework for analyzing long-time propagation of chaos in stochastic mean field systems, including path-dependent cases.

Findings

Propagation of chaos depends on initial $L^2$-Wasserstein distance.

Long-time entropy propagation is established under dissipative conditions.

Results apply even when log-Sobolev inequality does not hold.

Abstract

Due to the regularization effect of the stochastic noise, the quantitative entropy-cost type propagation of chaos for mean field interacting particle system is proposed. The result shows that the Kac's chaotic property measured in relative entropy at any positive time can only depend on the weaker initial one measured in $L^2$-Wasserstein distance. Moreover, under dissipative assumption, the long time entropy-cost type propagation of chaos can also be captured. The results are also available in path dependent case, where the log-Sobolev inequality for McKean-Vlasov SDEs does not hold.