Let @math be a symmetric probability measure of finite entropy on a group @math . We show that if @math , then the pair @math has the Liouville property (all bounded @math -harmonic functions on @math are constant). Furthermore, if @math where @math , then the entropy of the @math -fold convolution power @math satisfies @math . This improves earlier results of Gournay and of Saloff-Coste and the second author. We extend the bounds to transitive graphs and illustrate their sharpness on a family of groups.
In the work of the second author with Saloff-Coste @cite_12 , an upper bound similar to Theorem was proved. More precisely, suppose @math is a symmetric probability measure of finite second moment on @math such that @math for some @math , then by [Theorem 1.7] Saloff-Coste2014 [ H_ (n) (n ^ 1+ n )^ 1- for any >0. ] In particular, this bound implies the Liouville property of @math for @math . Our results in this paper imply a sharper upper bound @math under the same assumption, and are applicable to a larger class of decay lower bounds.
@cite_12
1981891714
Let @math and @math be standard normal random variables with covariance matrices @math and @math , respectively. Slepian's lemma says that if @math for @math , the lower bound @math is at least @math . In this paper an upper bound is given. The usefulness of the upper bound is justified with three concrete applications: (i) the new law of the iterated logarithm of Erdős and Revesz, (ii) the probability that a random polynomial does not have a real zero and (iii) the random pursuit problem for fractional Brownian particles. In particular, a conjecture of Kesten (1992) on the random pursuit problem for Brownian particles is confirmed, which leads to estimates of principal eigenvalues.