Central limit theorem for superdiffusive reflected Brownian motion

TL;DR

This paper establishes a central limit theorem for multidimensional reflected Brownian motion in complex domains, revealing Gaussian limits, phase transitions, and asymptotic independence of components.

Contribution

It provides the first detailed second-order asymptotics and CLT results for superdiffusive reflected Brownian motion in generalised parabolic domains.

Findings

Gaussian limit in unbounded direction with diffusive scaling

Convergence to invariant law in cross-sectional slice

Phase transition where CLT fails in narrow domains

Abstract

We study the second-order asymptotics around the superdiffusive strong law~\cite{MMW} of a multidimensional driftless diffusion with oblique reflection from the boundary in a generalised parabolic domain. In the unbounded direction we prove the limit is Gaussian with the usual diffusive scaling, while in the appropriately scaled cross-sectional slice we establish convergence to the invariant law of a reflecting diffusion in a unit ball. Using the separation of time scales, we also show asymptotic independence between these two components. The parameters of the limit laws are explicit in the growth rate of the boundary and the asymptotic diffusion matrix and reflection vector field. A phase transition occurs when the domain becomes too narrow, in which case we prove that the central limit theorem for the unbounded component fails.