多维周期扩散过程占位测度的Donsker定理
Donsker Theorems for Occupation Measures of Multi-Dimensional Periodic Diffusions
AI总结：
本文研究多维周期扩散过程经验过程的Donsker性质，利用生成元平滑性证明光滑函数类满足Donsker定理，并推广一维情形结论，进而得到d≤3时占位测度与不变测度间Wasserstein-1距离的精确渐近。
摘要：
我们研究由具有周期漂移和扩散系数的多维扩散过程产生的经验过程。利用扩散过程生成元的平滑性质，证明了某些光滑函数类的Donsker性质。我们部分推广了[van der Vaart & van Zanten, 2005]中研究的一维情形的结论：扩散经验过程比经典独立同分布观测情形表现出更强的正则性。作为一个应用，在维数$d\leq3$时，推导了时间-$T$占位测度与不变测度之间Wasserstein-1距离的精确渐近性质。
英文摘要：
We study the empirical process arising from a multi-dimensional diffusion process with periodic drift and diffusivity. The smoothing properties of the generator of the diffusion are exploited to prove the Donsker property for certain classes of smooth functions. We partially generalise the finding from the one-dimensional case studied in [van der Vaart & van Zanten, 2005]: that the diffusion empirical process exhibits stronger regularity than in the classical case of i.i.d. observations. As an application, precise asymptotics are deduced for the Wasserstein-1 distance between the time-$T$ occupation measure and the invariant measure in dimensions $d\leq3$.
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