Precise Limit in Wasserstein Distance for Conditional Empirical Measures of Dirichlet Diffusion Processes

TL;DR

This paper establishes a precise asymptotic limit for the Wasserstein distance between the conditional empirical measure of a Dirichlet diffusion process and its invariant measure, revealing detailed spectral dependence.

Contribution

It provides an explicit limit formula for the Wasserstein distance in Dirichlet diffusion processes, connecting spectral properties to empirical measure convergence.

Findings

Explicit limit formula involving eigenvalues and eigenfunctions.

Spectral gap influences the convergence rate.

Results applicable to compact Riemannian manifolds with boundary.

Abstract

Let $M$ be a $d$-dimensional connected compact Riemannian manifold with boundary $\partial M$, let $V\in C^2(M)$ such that $\mu(dx):=e^{V(x)} d x$ is a probability measure, and let $X_t$ be the diffusion process generated by $L:=\Delta+\nabla V$ with $\tau:=\inf\{t\ge 0: X_t\in\partial M\}$. Consider the conditional empirical measure $\mu_t^\nu:= \mathbb E^\nu\big(\frac 1 t \int_0^t \delta_{X_s}d s\big|t<\tau\big)$ for the diffusion process with initial distribution $\nu$ such that $\nu(\partial M)<1$. Then $$\lim_{t\to\infty} \big\{t\mathbb W_2(\mu_t^\nu,\mu_0)\big\}^2 = \frac 1 {\{\mu(\phi_0)\nu(\phi_0)\}^2} \sum_{m=1}^\infty \frac{\{\nu(\phi_0)\mu(\phi_m)+ \mu(\phi_0) \nu(\phi_m)\}^2}{(\lambda_m-\lambda_0)^3},$$ where $\nu(f):=\int_Mf {d} \nu$ for a measure $\nu$ and $f\in L^1(\nu)$, $\mu_0:=\phi_0^2\mu$, $\{\phi_m\}_{m\ge 0}$ is the eigenbasis of $-L$ in $L^2(\mu)$ with the Dirichlet boundary, $\{\lambda_m\}_{m\ge 0}$ are the corresponding Dirichlet eigenvalues, and $\mathbb W_2$ is the $L^2$-Wasserstein distance induced by the Riemannian metric.