Donsker Theorems for Occupation Measures of Multi-Dimensional Periodic Diffusions

TL;DR

This paper establishes Donsker theorems for occupation measures of multi-dimensional periodic diffusions, revealing enhanced regularity properties and providing asymptotic results for Wasserstein distances in low dimensions.

Contribution

It generalizes one-dimensional diffusion empirical process results to higher dimensions, demonstrating stronger regularity and deriving asymptotics for Wasserstein-1 distances.

Findings

Proves Donsker property for certain classes of smooth functions in multi-dimensional diffusions.

Shows the diffusion empirical process has stronger regularity than i.i.d. cases.

Provides asymptotic behavior of Wasserstein-1 distance between occupation and invariant measures in dimensions d≤3.

Abstract

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 $d\leq3$.