Curvature bound of Dyson Brownian Motion

Abstract

We construct a strongly local symmetric Dirichlet form on the configuration space ${\rm Y}$ 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-Émery gradient estimate $\mathsf{BE}(K,\infty )$ 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 ${\bar{\mathsf{d}}}_{{\rm Y}}$ on ${\rm Y}$, 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 ${\bar{\mathsf{d}}}_{{\rm Y}}$. As a result of $\mathsf{BE}(0,\infty )$, we obtain that the dual semigroup on the space of probability measures over ${\rm Y}$, endowed with a Benamou--Brenier-like extended distance ${\mathsf{W}}_{\mathcal{E}}$, satisfies the evolutional variation inequality with respect to the Bolzmann--Shannon entropy ${\mathsf{Ent}}_{{\mathsf{sine}}_{\beta }}$ 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 $0<s<1$.

55 pages. Final version

References in corpus (15)

• Metric measure spaces with Riemannian Ricci curvature bounded from below
• Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below
• Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds
• Continuum limits of random matrices and the Brownian carousel
• Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in RCD(K,\infty) metric measure spaces
• Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials
• Non-Equilibrium Dynamics of Dyson's Model with an Infinite Number of Particles
• On quotients of spaces with Ricci curvature bounded below
• Infinite Dimensional Stochastic Differential Equations for Dyson's Model
• Level statistics for one-dimensional Schrödinger operators and Gaussian beta ensemble
• Rademacher-type Theorems and Sobolev-to-Lipschitz Properties for Strongly Local Dirichlet Spaces
• Functional inequalities for the heat flow on time-dependent metric measure spaces
• Infinite-dimensional stochastic differential equations and tail $\sigma$-fields II: the IFC condition
• Ergodicity of unlabeled dynamics of Dyson's model in infinite dimensions
• The Infinite Dyson Brownian Motion with $\beta =2$ Does Not Have a Spectral Gap

Cited by in corpus (1)

• BV Functions and Sets of Finite Perimeter on Configuration Spaces