Precise estimates on the rate at which certain diffusions tend to equilibrium
type: Item
instance of: scholarly article
title: Precise estimates on the rate at which certain diffusions tend to equilibrium
zbMATH Open document ID: 0815.60074
DOI: 10.1007/BF02571965
author: Laurent Saloff-Coste
published in: Mathematische Zeitschrift
publication date: 1995-03-27T00:00:00Z
full work available at URL: https://eudml.org/doc/174711
review text: We consider a number of specific examples of ergodic self-adjoint Markov semigroups for which we obtain quantitative bounds on the time needed to reach equilibrium. For instance, we show that the time to equilibrium of the heat diffusion on the \(n\)-dimensional sphere is asymptotic to \({1\over 2}\log n\). Similar results are obtained for all rank one compact symmetric spaces and certain semigroups associated with Jacobi polynomials. Semi-simple connected compact Lie groups are also considered. In this case, the upper bound follows from precise Sobolev and logarithmic Sobolev inequalities whereas the lower bound uses representation theory. We show that the time to equilibrium and the logarithmic Sobolev constant \(\alpha\) are strongly related. This shed some new light on the log-Sobolev constant. In particular, we obtain upper bounds on \(\alpha\) that improve upon the universal estimate \(\alpha \leq \lambda/2\). Here, \(\lambda\) is the spectral gap. For rank one compact symmetric spaces, we show that the equality \(\alpha = \lambda/2\) characterizes the spheres. Moreover, for the projective spaces, we show that \(\alpha \sim \lambda/4\) as the dimension tends to infinity.
Mathematics Subject Classification ID: 60J35
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 47D07
zbMATH DE Number: 737158
zbMATH Keywords: semi-simple connected compact Lie groups
zbMATH Keywords: ergodic self-adjoint Markov semigroups
zbMATH Keywords: heat diffusion
zbMATH Keywords: logarithmic Sobolev inequalities
MaRDI profile type: Publication
cites work: Strong uniform times and finite random walks
cites work: Q5580319
cites work: Q3704110
cites work: On the spectrum of non-compact manifolds with finite volume
cites work: Upper bounds for symmetric Markov transition functions
cites work: Q4041031
cites work: Q4692653
cites work: Q3995195
cites work: Generating a random permutation with random transpositions
cites work: Q3711390
cites work: Logarithmic Sobolev inequalities for finite Markov chains
cites work: Q4002278
cites work: Q5539480
cites work: Linearization of the Product of Jacobi Polynomials. I
cites work: Logarithmic Sobolev Inequalities
cites work: Q3331943
cites work: Uniform and \(L^ 2\) convergence in one dimensional stochastic Ising models
cites work: Constantes explicites pour les inégalités de Sobolev sur les variétés Riemanniennes compactes
cites work: An Example in the Theory of Hypercontractive Semigroups
cites work: On an integral criterion for hypercontractivity of diffusion semigroups and extremal functions
cites work: Hypercontractivity for the heat semigroup for ultraspherical polynomials and on the n-sphere
cites work: Continuity of Solutions of Parabolic and Elliptic Equations
cites work: Q3998356
cites work: Random rotations: Characters and random walks on \(SO(N)\)
cites work: Diffusion on compact Riemannian manifolds and logarithmic Sobolev inequalities
cites work: Hypercontractivity and the Bakry-Emery criterion for compact Lie groups
cites work: Convergence to equilibrium and logarithmic Sobolev constant on manifolds with Ricci curvature bounded below
cites work: A Remark on Nelson's Best Hypercontractive Estimates
cites work: Q5640160
cites work: The equivalence of the logarithmic Sobolev inequality and the Dobrushin- Shlosman mixing condition
cites work: Q3681117
cites work: Q3607418
Wikidata QID: Q125774023
description: scientific article; zbMATH DE number 737158