Log-unimodality for free positive multiplicative Brownian motion

TL;DR

This paper proves that the distribution of free positive multiplicative Brownian motion maintains log-unimodality under certain conditions, with specific results for symmetric and bounded support cases, and provides counterexamples otherwise.

Contribution

It establishes log-unimodality of the marginal law of free positive multiplicative Brownian motion under symmetry and support conditions, extending understanding of its distributional properties.

Findings

Log-unimodality holds for symmetric, log-unimodal initial distributions.

Log-unimodality holds for large t with bounded support initial distributions.

Counterexamples show failure without symmetry or bounded support.

Abstract

We prove that the marginal law $\sigma_{t}\boxtimes\nu$ of free positive multiplicative Brownian motion is log-unimodal for all $t>0$ if $\nu$ is a multiplicatively symmetric log-unimodal distribution, and that $\sigma_{t}\boxtimes\nu$ is log-unimodal for sufficiently large $t$ if $\nu$ is supported on a suitably chosen finite interval. Counterexamples are given when $\nu$ is not assumed to be symmetric or having a bounded support.