Curvature bound of Dyson Brownian Motion

Kohei Suzuki

Published 2022-12-31Version 1

In this article, we show $1$-Bakry-\'Emery lower Ricci curvature bound $\mathrm{BE}_1(0, \infty)$ of a Dirichlet form on the configuration space whose invariant measure is $\mathsf{sine}_\beta$ ensemble for any $\beta>0$. As a particular case of $\beta=2$, our result proves $\mathrm{BE}_1(0, \infty)$ for a Dirichlet form related to the unlablled Dyson Brownian motion. We prove furthermore several functional inequalities including the integral Bochner inequality, the local Poincar\'e and the local log-Sobolev inequalities as well as the log-Harnack and the dimension-free Harnack inequalities, the Lipschitz contraction property and the $L^\infty$-to-Lipschitz regularisation property of the semigroup with the $L^2$-transportation-type extended distance. At the end of the article, we provide a sufficient condition for the synthetic lower Ricci curvature bound in the case of general invariant measures beyond~$\mathsf{sine}_\beta$.