Extension of a theorem of Wschebor to free and matrix Brownian motions

TL;DR

This paper extends Wschebor's 1992 theorem on Brownian motion increments to Hermitian and free Brownian motions, also analyzing the convergence of fluctuations to a deterministic limit.

Contribution

It introduces a new version of Wschebor's theorem applicable to free and matrix Brownian motions, including fluctuation convergence analysis.

Findings

Proves convergence of small increments for Hermitian Brownian motion

Establishes convergence in distribution of fluctuations for free Brownian motion

Extends classical theorem to non-commutative stochastic processes

Abstract

In 1992, M. Wschebor proved a theorem on the convergence of small increments of the Brownian motion. Since then, it has been extended to various processes. We prove a version of this theorem for the Hermitian Brownian motion and the free Brownian motion. Since these theorems deal with a convergence to a deterministic limit, we prove also the convergence in distribution of the corresponding fluctuations.