Strong and weak convergence for averaging principle of DDSDE with singular drift

TL;DR

This paper investigates the averaging principle for distribution-dependent stochastic differential equations with singular drifts, establishing strong and weak convergence results and their rates as the time scale approaches zero.

Contribution

It introduces a novel approach using Zvonkin's transformation to prove convergence for DDSDEs with singular drifts, providing explicit convergence rates.

Findings

Solutions converge strongly and weakly to the averaged system as epsilon approaches zero.

Convergence rates depend explicitly on the integrability parameter p.

The method extends averaging principles to systems with singular drifts in localized L^p spaces.

Abstract

In this paper, we study the averaging principle for distribution dependent stochastic differential equations with drift in localized $L^p$ spaces. Using Zvonkin's transformation and estimates for solutions to Kolmogorov equations, we prove that the solutions of the original system strongly and weakly converge to the solution of the averaged system as the time scale $\eps$ goes to zero. Moreover, we obtain rates of the strong and weak convergence that depend on $p$ respectively.