On Groups, Slow Heat Kernel Decay Yields Liouville Property and Sharp Entropy Bounds

摘要

Let $mu $ be a symmetric probability measure of finite entropy on a group G. We show that if $-log mu ^{(2n)}(extrm{id})=oleft(n^{1/2}ight)$, then the pair $(G,mu )$ has the Liouville property (all bounded $mu $-harmonic functions on G are constant). Furthermore, if $-log mu ^{(2n)}(extrm{id})=Oleft(n^{eta }ight)$ where $eta in (0,1/2)$, then the entropy of the n-fold convolution power $mu ^{(n)}$ satisfies $Hleft(mu ^{(n)}ight)=OBig (n^{rac{eta }{1-eta }}Big )$. These results improve earlier work of Gournay [16] and Saloff-Coste and Zheng [27]. We illustrate the sharpness of the bounds on a family of groups.