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Convergence in Wasserstein distance for empirical measures of Dirichlet diffusion processes on manifolds

Feng-yu Wang Orcid Logo

Journal of the European Mathematical Society, Volume: 25, Issue: 9

Swansea University Author: Feng-yu Wang Orcid Logo

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DOI (Published version): 10.4171/jems/1269

Abstract

Let M be a d-dimensional connected compact Riemannian manifold with boundary ∂M, let V∈C2(M) such that μ(dx):=eV(x)dx is a probability measure, and let Xt be the diffusion process generated by L:=Δ+∇V with τ:=inf{t≥0:Xt∈∂M}. Consider the empirical measure μt:=1t∫t0δXsds under the condition t<τ fo...

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Published in: Journal of the European Mathematical Society
ISSN: 1435-9855 1435-9863
Published: European Mathematical Society - EMS - Publishing House GmbH 2022
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URI: https://cronfa.swan.ac.uk/Record/cronfa60533
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Abstract: Let M be a d-dimensional connected compact Riemannian manifold with boundary ∂M, let V∈C2(M) such that μ(dx):=eV(x)dx is a probability measure, and let Xt be the diffusion process generated by L:=Δ+∇V with τ:=inf{t≥0:Xt∈∂M}. Consider the empirical measure μt:=1t∫t0δXsds under the condition t<τ for the diffusion process. If d≤3, then for any initial distribution not fully supported on ∂M,c∑m=1∞2(λm−λ0)2≤lim inft→∞infT≥t{tE[W2(μt,μ0)2∣∣T<τ]}≤lim supt→∞supT≥t{tE[W2(μt,μ0)2∣∣T<τ]}≤∑m=1∞2(λm−λ0)2holds for some constant c∈(0,1] with c=1 when ∂M is convex, where μ0:=ϕ20μ for the first Dirichet eigenfunction ϕ0 of L, {λm}m≥0 are the Dirichlet eigenvalues of −L listed in the increasing order counting multiplicities, and the upper bound is finite if and only if d≤3. When d=4, supT≥tE[W2(μt,μ0)2∣∣T<τ] decays in the order t−1logt, while for d≥5 it behaves like t−2d−2, as t→∞.
Keywords: Conditional empirical measure, Dirichlet diffusion process, Wasserstein distance, eigenvalues, eigenfunctions.
College: Faculty of Science and Engineering
Funders: Supported in part by the National Key R&D Program of China (No. 2020YFA0712900) and NNSFC (11831014, 11921001).
Issue: 9

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