Uniform-in-time estimates for mean-field type SDEs and applications

TL;DR

This paper develops uniform-in-time probability distance estimates for mean-field SDEs with dissipative drifts, enabling analysis of long-term behavior, propagation of chaos, and discretization errors in stochastic algorithms.

Contribution

It introduces an asymptotic reflection coupling method to establish uniform-in-time estimates for mean-field SDEs with dissipative drifts, including singular and non-convolution cases.

Findings

Established uniform-in-time probability distance bounds for mean-field SDEs.

Analyzed long-term distance between SDEs and their delay versions.

Provided discretization error bounds for stochastic algorithms over infinite time.

Abstract

Via constructing an asymptotic coupling by reflection, in this paper we establish uniform-in-time estimates on probability distances for mean-field type SDEs, where the drift terms under consideration are dissipative merely in the long distance. As applications, we (i) explore the long time probability distance estimate between an SDE and its delay version; (ii) investigate the issue on uniform-in-time propagation of chaos for McKean-Vlasov SDEs, where the drifts might be singular with respect to the spatial variables and need not to be of convolution type; (iii) tackle the discretization error bounds in an infinite-time horizon for stochastic algorithms (e.g. backward/tamed/adaptive Euler-Maruyama schemes as three typical candidates) associated with McKean-Vlasov SDEs.