On the convergence to equilibrium of Brownian motion on compact simple Lie groups (Q1768090)

Let \(M\) be a compact Riemannian manifold and \(h(t,x, y)\) be the symmetric kernel associated to the heat diffusion semigroup on \(L^2(M)\). For any \(p\geq 1\), one defines \(T_p(\varepsilon ) = \inf \{ t >0 \mid \sup _{x\in M} \| h(t,x, \cdot )-1\| _p \leq \epsilon \}\). In this article the author obtains lower and upper bounds for \(T_p(\varepsilon )\) when \(M\) is a compact connected simple Lie group or when \(M\) is an irreducible compact symmetric space. This work is a continuation of the author's previous works [Math. Z. 217, 641--677 (1994; Zbl 0815.60074) and Colloq. Math. 67, 109--121 (1994; Zbl 0816.53027)]. For general compact manifolds, a lower bound is obtained and many examples are discussed to show the use of this lower bound.