论文标题
Wasserstein渐近学用于平面圆环的分数布朗运动的经验度量
Wasserstein Asymptotics for the Empirical Measure of Fractional Brownian Motion on a Flat Torus
论文作者
论文摘要
我们在任何阶段的wasserstein距离上建立了渐近的上限和下限，$ p \ ge 1 $在平坦的圆环上的分数布朗运动和统一的lebesgue措施之间。我们的不平等表明，赫斯特指数$ h $与国家空间的尺寸$ d $之间的有趣互动，当$ d = 2+1/h $，类似于ajtai-komlós-tusnádytusnádytheorem the速率时，速率转移了。二维点。我们的证明夫妻PDE和概率技术，并且对于离散时间近似过程，以及在$ \ m athbb {r}^d $上的同一问题的下限也产生了相似的结果。
We establish asymptotic upper and lower bounds for the Wasserstein distance of any order $p\ge 1$ between the empirical measure of a fractional Brownian motion on a flat torus and the uniform Lebesgue measure. Our inequalities reveal an interesting interaction between the Hurst index $H$ and the dimension $d$ of the state space, with a "phase-transition" in the rates when $d=2+1/H$, akin to the Ajtai-Komlós-Tusnády theorem for the optimal matching of i.i.d. points in two-dimensions. Our proof couples PDE's and probabilistic techniques, and also yields a similar result for discrete-time approximations of the process, as well as a lower bound for the same problem on $\mathbb{R}^d$.
