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Log-unimodality for free positive multiplicative Brownian motion

Volume 169 / 2022

Takahiro Hasebe, Yuki Ueda, Jiun-Chau Wang Colloquium Mathematicum 169 (2022), 209-226 MSC: Primary 46L54; Secondary 60J65, 60E07, 60B15. DOI: 10.4064/cm8413-6-2021 Published online: 22 February 2022


We prove that the marginal law $\sigma _{t}\mathrel {\scriptstyle {\boxtimes }}\nu $ of free positive multiplicative Brow\-nian motion is log-unimodal for all $t \gt 0$ if $\nu $ is a multiplicatively symmetric log-unimodal distribution, and that $\sigma _{t}\mathrel {\scriptstyle {\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.


  • Takahiro HasebeDepartment of Mathematics
    Hokkaido University
    Kita 10, Nishi 8, Kita-Ku
    Sapporo, 060-0810, Hokkaido, Japan
  • Yuki UedaDepartment of General Science
    National Institute of Technology
    Ichinoseki College Takanashi, Hagisho
    Ichinoseki, 021-8511, Iwate, Japan
    Department of Mathematics
    Hokkaido University of Education
    9 Hokumon-cho
    Asahikawa, 070-8621, Hokkaido, Japan
  • Jiun-Chau WangDepartment of Mathematics and Statistics
    University of Saskatchewan
    106 Wiggins Road, Saskatoon
    S7N 5E6, Saskatchewan, Canada

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