2022/12/31 by Suzuki, Kohei · 2 citations
#Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Probability (math.PR)
paper · doi:10.48550/arxiv.2301.00262
We construct a strongly local symmetric Dirichlet form on the configuration space Υ whose symmetrising (thus also invariant) measure is sineβ, which is the law of the sine β ensemble for every β>0. For every β>0, this Dirichlet form satisfies the Bakry-Émery gradient estimate BE(K, ∞) 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 L2-transportation-type extended distance \sf dΥ on Υ, and prove the dimension-free Harnack inequality and several Lipschitz regularisation estimates of the L2-semigroup associated with the Dirichlet form in terms of \sf dΥ. As a result of BE(0,∞), we obtain that the dual semigroup on the space of probability measures over Υ, endowed with a Benamou--Brenier-like extended distance W\mathcal E, satisfies the evolutional variation inequality with respect to the Bolzmann--Shannon entropy Entsineβ associated with sineβ. Furthermore, the dual semigroup is characterised as the unique W\mathcal E-gradient flow in the space of probability measures with respect to Entsineβ. Finally, we provide a sufficient condition for BE(K, ∞) beyond sineβ and apply it to the infinite particle diffusion whose symmetrising measure is the law of the 1-dimensional (β,s)-circular Riesz gas with β>0 and 0