论文标题
瓦瑟尔斯坦距离的收敛，用于歧管上的迪利奇扩散过程的经验度量
Convergence in Wasserstein Distance for Empirical Measures of Dirichlet Diffusion Processes on Manifolds
论文作者
论文摘要
令$ m $为$ d $二维连接的紧凑型riemannian带有边界$ \ partial m $，让$ v \ in c^2（m）$中的$ v \这样，以至于$μ（{\ rm d} x）：= = {\ rm e}^{v（x） $ l：=δ+\ nabla v $ with $τ：= \ inf \ {t \ ge 0：x_t \ in \ partial m \} $。考虑经验度量$μ_t：= \ frac 1 t \ int_0^tΔ__{x_s} {\ rm d} s $在扩散过程的条件$ t <τ$下。如果$ d \ le 3 $，则对于任何初始分布，$ \ partial m $，\ begin {align*}＆c \ sum_ {m = 1}^\ infty \ frac {2} {（λ_m-λ_0）^2} \ big \ { e \ big [\ mathbb w_2（μ_t，μ_0）^2 \ big | t <τ\ big] $c=1$ when $\partial M$ is convex, where $μ_0:= ϕ_0^2μ$ for the first Dirichet eigenfunction $ϕ_0$ of $L$, $\{λ_m\}_{m\ge 0}$ are the Dirichlet eigenvalues of $-L$ listed in the increasing order counting multiplicities, and the upper bound is finite如果并且仅当$ d \ le 3 $。 $ t^{ - \ frac 2 {d-2}} $，as $ t \ to \ infty $。
Let $M$ be a $d$-dimensional connected compact Riemannian manifold with boundary $\partial M$, let $V\in C^2(M)$ such that $μ({\rm d} x):={\rm e}^{V(x)}{\rm d} x$ is a probability measure, and let $X_t$ be the diffusion process generated by $L:=Δ+\nabla V$ with $τ:=\inf\{t\ge 0: X_t\in\partial M\}$. Consider the empirical measure $μ_t:=\frac 1 t \int_0^t δ_{X_s}{\rm d} s$ under the condition $t<τ$ for the diffusion process. If $d\le 3$, then for any initial distribution not fully supported on $\partial M$, \begin{align*} &c\sum_{m=1}^\infty \frac{2}{(λ_m-λ_0)^2} \le \liminf_{t\to \infty} \inf_{T\ge t} \Big\{t {\mathbb E}\big[\mathbb W_2(μ_t, μ_0)^2\big|T<τ\big]\Big\} \\ &\le \limsup_{t\to \infty} \sup_{T\ge t} \Big\{ t \mathbb E\big[\mathbb W_2(μ_t, μ_0)^2\big|T<τ\big] \Big\}\le \sum_{m=1}^\infty \frac{2}{(λ_m-λ_0)^2}\end{align*} holds for some constant $c\in (0,1]$ with $c=1$ when $\partial M$ is convex, where $μ_0:= ϕ_0^2μ$ for the first Dirichet eigenfunction $ϕ_0$ of $L$, $\{λ_m\}_{m\ge 0}$ are the Dirichlet eigenvalues of $-L$ listed in the increasing order counting multiplicities, and the upper bound is finite if and only if $d\le 3$. When $d=4$, $\sup_{T\ge t} \mathbb E\big[\mathbb W_2(μ_t, μ_0)^2\big|T<τ\big] $ decays in the order $t^{-1}\log t$, while for $d\ge 5$ it behaves like $t^{-\frac 2 {d-2}}$, as $t\to\infty$.
