Journal article 557 views 71 downloads
Convergence in Wasserstein distance for empirical measures of Dirichlet diffusion processes on manifolds
Feng-yu Wang
Journal of the European Mathematical Society, Volume: 25, Issue: 9
Swansea University Author: Feng-yu Wang
-
PDF | Accepted Manuscript
Download (364.88KB)
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...
| Published in: | Journal of the European Mathematical Society |
|---|---|
| ISSN: | 1435-9855 1435-9863 |
| Published: |
European Mathematical Society - EMS - Publishing House GmbH
2022
|
| Online Access: |
Check full text
|
| URI: | https://cronfa.swan.ac.uk/Record/cronfa60533 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| 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 |