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Precise limit in Wasserstein distance for conditional empirical measures of Dirichlet diffusion processes
Journal of Functional Analysis ( IF 1.7 ) Pub Date : 2021-03-16 , DOI: 10.1016/j.jfa.2021.108998
Feng-Yu Wang

Let M be a d-dimensional connected compact Riemannian manifold with boundary ∂M, let VC2(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{t0:XtM}. Consider the conditional empirical measure μtν:=Eν(1t0tδXsds|t<τ) for the diffusion process with initial distribution ν such that ν(M)<1. Thenlimt{tW2(μtν,μ0)}2=1{μ(ϕ0)ν(ϕ0)}2m=1{ν(ϕ0)μ(ϕm)+μ(ϕ0)ν(ϕm)}2(λmλ0)3, where ν(f):=Mfdν for a measure ν and fL1(ν), μ0:=ϕ02μ, {ϕm}m0 is the eigenbasis of −L in L2(μ) with the Dirichlet boundary, {λm}m0 are the corresponding Dirichlet eigenvalues, and W2 is the L2-Wasserstein distance induced by the Riemannian metric.



中文翻译:

Dirichlet扩散过程的条件经验测度的Wasserstein距离精确极限

中号d维连接的紧黎曼流形与边界∂中号,让伏特C2个中号 这样 μdX=Ë伏特XdX 是一种概率测度, XŤ 是由产生的扩散过程 大号=Δ+伏特τ=信息{Ť0XŤ中号}。考虑有条件的经验测度μŤν=Eν1个Ť0ŤδXsds|Ť<τ对于具有初始分布ν的扩散过程,使得ν中号<1个。然后Ť{Ťw ^2个μŤνμ0}2个=1个{μϕ0νϕ0}2个=1个{νϕ0μϕ+μϕ0νϕ}2个λ-λ03 在哪里 νF=中号Fdν测度νF大号1个νμ0=ϕ02个μ{ϕ}0-L in的本征基大号2个μ 与狄利克雷边界, {λ}0 是对应的Dirichlet特征值,并且 w ^2个 是个 大号2个黎曼度量引起的-Wasserstein距离。

更新日期:2021-03-16
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