Precise estimates on the rate at which certain diffusions tend to equilibrium
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.
semi-simple connected compact Lie groups
scientific article; zbMATH DE number 737158
scientific article