论文标题
Dirichlet扩散过程的有条件经验度量的Wasserstein距离的精确极限
Precise Limit in Wasserstein Distance for Conditional Empirical Measures of Dirichlet Diffusion Processes
论文作者
论文摘要
令$ m $为$ d $二维连接的紧凑型riemannian歧管，带有边界$ \ partial m $，让$ v \ in c^2（m）$中的$ v \，使$μ（dx）：= = e^{v（x）d x $是概率度量，让$ x_t $是$ x_t $ by $ x_t $ by $ l： $τ：= \ inf \ {t \ ge 0：x_t \ in \ partial m \} $。考虑条件经验措施 $μ_t^ν：= \ MATHBB E^ν\ big（\ frac 1 t \ int_0^tδ_{x_s} d s \ big | t <| t <τ\ big）$，用于初始分布过程$ν$，因此然后$ \ lim_ {t \ to \ infty} \ big \ { \ frac {\ {ν（ϕ_0）μ（ϕ_m）+μ（ϕ_0）ν（ϕ_m）\}^2} {（λ_m-λ_0）^3}，$ n $ n $ n $ = = = \ int_mf {d} $ $μ_0：= ϕ_0^2μ$，$ \ {ϕ_m \} _ {m \ ge 0} $是$ -l $ in $ l^2（μ）$ in dirichlet边界，$ -l $ in $ -l $， $ \ {λ_m\} _ {m \ ge 0} $是相应的dirichlet eigenvalues，而$ \ mathbb w_2 $是$ l^2 $ -WASSERSTEIN距离，由Riemannian Metric引起的距离。
Let $M$ be a $d$-dimensional connected compact Riemannian manifold with boundary $\partial M$, let $V\in C^2(M)$ such that $μ(dx):=e^{V(x)} d x$ 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 conditional empirical measure $μ_t^ν:= \mathbb E^ν\big(\frac 1 t \int_0^t δ_{X_s}d s\big|t<τ\big)$ for the diffusion process with initial distribution $ν$ such that $ν(\partial M)<1$. Then $$\lim_{t\to\infty} \big\{t\mathbb W_2(μ_t^ν,μ_0)\big\}^2 = \frac 1 {\{μ(ϕ_0)ν(ϕ_0)\}^2} \sum_{m=1}^\infty \frac{\{ν(ϕ_0)μ(ϕ_m)+ μ(ϕ_0) ν(ϕ_m)\}^2}{(λ_m-λ_0)^3},$$ where $ν(f):=\int_Mf {d} ν$ for a measure $ν$ and $f\in L^1(ν)$, $μ_0:=ϕ_0^2μ$, $\{ϕ_m\}_{m\ge 0}$ is the eigenbasis of $-L$ in $L^2(μ)$ with the Dirichlet boundary, $\{λ_m\}_{m\ge 0}$ are the corresponding Dirichlet eigenvalues, and $\mathbb W_2$ is the $L^2$-Wasserstein distance induced by the Riemannian metric.
