Convergence Rate of Particle System for Second-order PDEs On Wasserstein Space

TL;DR

The article proves a convergence rate for particle approximations of a class of second-order PDEs on the Wasserstein space, establishing that the infinite-dimensional inf/sup-convolution of the finite-particle value yields a viscosity supersolution up to an error term and enabling a rate via a comparison principle. The main result shows $\sup_{(t,\boldsymbol{x})}|v^N(t,\boldsymbol{x})-v(t,\mu^{\boldsymbol{x}})| \le C\alpha^{1/3}(N)$ with $\alpha(N)$ depending on the dimension $d$, by choosing the convolution parameter $\epsilon=\alpha(N)$ and exploiting regularity from prior work. The approach is analytic and does not rely on semi-concavity, and it provides the first convergence rate for particle approximations of partially second-order PDEs on the Wasserstein space, with applications to mean-field control with common noise. The study integrates viscosity theory on Wasserstein space, Lions differentiability, and Fourier-Wasserstein metrics to connect finite-particle dynamics to the mean-field limit.

Abstract

In this paper, we provide a convergence rate for particle approximations of a class of second-order PDEs on Wasserstein space. We show that, up to some error term, the infinite-dimensional inf(sup)-convolution of the finite-dimensional value function yields a super (sub)-viscosity solution to the PDEs on Wasserstein space. Hence, we obtain a convergence rate using a comparison principle of such PDEs on Wasserstein space. Our argument is purely analytic and relies on the regularity of value functions established in \cite{DaJaSe23}.

Theorems & Definitions (18)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Lemma 2.1
• Lemma 2.2
• proof
• Proposition 2.1
• proof
• Proposition 2.2
• Lemma 2.3