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

Abstract

Let $M$ be a $d$-dimensional connected compact Riemannian manifold with boundary $\partial M$, let $V\in C^2(M)$ such that $\mu (dx):=e^{V(x)}dx$ 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 ${\mu }_t^{\nu }:={\mathbb{E}}^{\nu }\left(\frac{1}{t}{\int }_0^t{\delta }_{X_s}ds\right|t<\tau )$ for the diffusion process with initial distribution $\nu$ such that $\nu (\partial M)<1$. Then $$\underset{t\to \infty }{\lim }\{t{\mathbb{W}}_2({\mu }_t^{\nu },{\mu }_0){\}}^2=\frac{1}{\{\mu ({\varphi }_0)\nu ({\varphi }_0){\}}^2}\sum _{m=1}^{\infty }\frac{\{\nu ({\varphi }_0)\mu ({\varphi }_m)+\mu ({\varphi }_0)\nu ({\varphi }_m){\}}^2}{({\lambda }_m-{\lambda }_0{)}^3},$$ where $\nu (f):={\int }_{M}fd\nu$ for a measure $\nu$ and $f\in L^1(\nu )$, ${\mu }_0:={\varphi }_0^2\mu$, $\{{\varphi }_m{\}}_{m\ge 0}$ 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.

20 pages

References in corpus (1)

• Convergence in Wasserstein Distance for Empirical Measures of Dirichlet Diffusion Processes on Manifolds