Rate of convergence in the Smoluchowski-Kramers approximation for mean-field stochastic differential equations

TL;DR

This paper analyzes the convergence rates of second-order mean-field stochastic differential equations to their zero-mass limit, using Malliavin calculus to quantify the approximation in various distances.

Contribution

It provides explicit convergence rates in the zero-mass limit for mean-field SDEs, a novel application of Malliavin calculus in this context.

Findings

Established explicit convergence rates in $L^p$-distances.

Derived convergence rates in total variation distance.

Applied techniques to position, velocity, and re-scaled velocity processes.

Abstract

In this paper we study a second-order mean-field stochastic differential systems describing the movement of a particle under the influence of a time-dependent force, a friction, a mean-field interaction and a space and time-dependent stochastic noise. Using techniques from Malliavin calculus, we establish explicit rates of convergence in the zero-mass limit (Smoluchowski-Kramers approximation) in the $L^p$-distances and in the total variation distance for the position process, the velocity process and a re-scaled velocity process to their corresponding limiting processes.