Large time unimodality for classical and free Brownian motions with initial distributions

TL;DR

This paper demonstrates that classical and free Brownian motions become unimodal over time under certain initial distribution conditions, with extensions to stable processes and discussions on strong unimodality in free convolution.

Contribution

It establishes large-time unimodality results for classical and free Brownian motions with initial distributions, including optimal assumptions and extensions to stable processes.

Findings

Classical and free Brownian motions are unimodal for large time under certain initial conditions.

Unimodality results extend to symmetric and positive stable processes.

Discussion on strong unimodality for free convolution.

Abstract

We prove that classical and free Brownian motions with initial distributions are unimodal for sufficiently large time, under some assumption on the initial distributions. The assumption is almost optimal in some sense. Similar results are shown for a symmetric stable process with index 1 and a positive stable process with index $1/2$. We also prove that free Brownian motion with initial symmetric unimodal distribution is unimodal, and discuss strong unimodality for free convolution.