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

Let M = G/H bean irreducible homogeneous compact manifold of dimension n equipped with its canonical Riemannian metric. Let lambda be the lowest nonzero eigenvalue of the Laplace operator Let A be the normalized Haar measure and mu(t) be the heat diffusion measure, i.e., the law of Brownian motion started at a fixed origin in M. We show that the total variation distance between At and A is not small for t << lambda(-1) log n. This is sharp, up to a factor of two, in the case of compact irreducible simply connected symmetric spaces.