Quantitative estimate of the overdamped limit for the Vlasov-Fokker-Planck systems

TL;DR

This paper provides a probabilistic method to quantitatively estimate the overdamped limit of the Vlasov-Fokker-Planck system, including cases with singular forces, by analyzing the weak convergence of associated stochastic differential equations.

Contribution

It introduces a probabilistic approach to quantify the overdamped limit for Vlasov-Fokker-Planck equations, extending previous results to more general force cases.

Findings

Established a convergence rate for the overdamped limit

Applied probabilistic techniques to analyze SDEs of McKean type

Included cases with Newtonian singular forces

Abstract

This note adapts a probabilistic approach to establish a quantified estimate of the overdamped limit for the Vlasov-Fokker-Planck equation towards the aggregation-diffusion equation, which in particular includes cases of the Newtonian type singular forces. The proofs are based on the investigation of the weak convergence of the corresponding stochastic differential equations (SDEs) of Mckean type in the continuous path space. We show that one can obtain the same convergence rate as in [10] under the same assumptions.