Curvature bound of Dyson Brownian Motion

We construct a strongly local symmetric Dirichlet form on the configuration space $\mathrm{{\rm Y}}$ whose symmetrising (thus also invariant) measure is ${\mathsf{s}\mathsf{i}\mathsf{n}\mathsf{e}}_{\beta }$, which is the law of the sine $\beta$ ensemble for every $\beta >0$. For every $\beta >0$, this Dirichlet form satisfies the Bakry-Émery gradient estimate $\mathsf{B}\mathsf{E}(K,\mathrm{\infty })$ with $K=0$. This implies various functional inequalities, including the local Poincaré inequality, the local log-Sobolev inequality and the local hyper-contractivity. We then introduce an $L^2$-transportation-type extended distance ${\bar{\mathsf{d}}}_{\mathrm{{\rm Y}}}$ on $\mathrm{{\rm Y}}$, 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 ${\bar{\mathsf{d}}}_{\mathrm{{\rm Y}}}$. As a result of $\mathsf{B}\mathsf{E}(0,\mathrm{\infty })$, we obtain that the dual semigroup on the space of probability measures over $\mathrm{{\rm Y}}$, endowed with a Benamou–Brenier-like extended distance ${\mathsf{W}}_{\mathcal{E}}$, satisfies the evolutional variation inequality with respect to the Bolzmann–Shannon entropy ${\mathsf{E}\mathsf{n}\mathsf{t}}_{{\mathsf{s}\mathsf{i}\mathsf{n}\mathsf{e}}_{\beta }}$ associated with ${\mathsf{s}\mathsf{i}\mathsf{n}\mathsf{e}}_{\beta }$. Furthermore, the dual semigroup is characterised as the unique ${\mathsf{W}}_{\mathcal{E}}$-gradient flow in the space of probability measures with respect to ${\mathsf{E}\mathsf{n}\mathsf{t}}_{{\mathsf{s}\mathsf{i}\mathsf{n}\mathsf{e}}_{\beta }}$. Finally, we provide a sufficient condition for $\mathsf{B}\mathsf{E}(K,\mathrm{\infty })$ beyond ${\mathsf{s}\mathsf{i}\mathsf{n}\mathsf{e}}_{\beta }$ 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 $0<s<1$.