Rate of convergence for particle approximation of PDEs in Wasserstein space

TL;DR

This paper establishes convergence rates for particle approximations of certain PDEs in Wasserstein space, specifically in the context of mean-field control problems with common noise, using backward SDE techniques.

Contribution

It provides the first known explicit convergence rates for particle methods approximating second-order PDEs in Wasserstein space related to mean-field control.

Findings

Rate of $1/N$ for pathwise error in solution $v$

Rate of $1/ oot{N}{}$ for $L^2$-error in $L$-derivative $\partial_\mu v$

Proof utilizes backward stochastic differential equations techniques

Abstract

We prove a rate of convergence for the $N$-particle approximation of a second-order partial differential equation in the space of probability measures, like the Master equation or Bellman equation of mean-field control problem under common noise. The rate is of order $1/N$ for the pathwise error on the solution $v$ and of order $1/\sqrt{N}$ for the $L^2$-error on its $L$-derivative $\partial_\mu v$. The proof relies on backward stochastic differential equations techniques.