Curvature bound of Dyson Brownian Motion

Abstract

We construct a strongly local symmetric Dirichlet form on the configuration space $\Upsilon$ whose symmetrising (thus also invariant) measure is $\mathsf{sine}_\beta$, which is the law of the sine $\beta$ ensemble for every $\beta>0$. For every $\beta>0$, this Dirichlet form satisfies the Bakry-\'Emery gradient estimate $\mathsf{BE}(K, \infty)$ with $K=0$. This implies various functional inequalities, including the local Poincar\'e inequality, the local log-Sobolev inequality and the local hyper-contractivity. We then introduce an $L^2$-transportation-type extended distance $\bar{\sf d}_{\Upsilon}$ on $\Upsilon$, and prove the dimension-free Harnack inequality and several Lipschitz regularisation estimates of the $L^2$-semigroup associated with the Dirichlet form in terms of $\bar{\sf d}_{\Upsilon}$. As a result of $\mathsf{BE}(0,\infty)$, we obtain that the dual semigroup on the space of probability measures over $\Upsilon$, endowed with a Benamou--Brenier-like extended distance $\mathsf{W}_{\mathcal E}$, satisfies the evolutional variation inequality with respect to the Bolzmann--Shannon entropy $\mathsf{Ent}_{\mathsf{sine}_\beta}$ associated with $\mathsf{sine}_\beta$. Furthermore, the dual semigroup is characterised as the unique $\mathsf{W}_{\mathcal E}$-gradient flow in the space of probability measures with respect to $\mathsf{Ent}_{\mathsf{sine}_\beta}$. Finally, we provide a sufficient condition for $\mathsf{BE}(K, \infty)$ beyond $\mathsf{sine}_\beta$ and apply it to the infinite particle diffusion whose symmetrising measure is the law of the $1$-dimensional $(\beta,s)$-circular Riesz gas with $\beta>0$ and $0<s<1$.