Continuity and estimates of the Liouville heat kernel with applications to spectral dimensions

Abstract

The Liouville Brownian motion (LBM), recently introduced by Garban, Rhodes and Vargas and in a weaker form also by Berestycki, is a diffusion process evolving in a planar random geometry induced by the Liouville measureMγ, formally written as Mγ(dz)=eγX(z)-γ2E[X(z)2]/2dz, γ∈ (0 , 2) , for a (massive) Gaussian free field X. It is an Mγ-symmetric diffusion defined as the time change of the two-dimensional Brownian motion by the positive continuous additive functional with Revuz measure Mγ. In this paper we provide a detailed analysis of the heat kernel pt(x, y) of the LBM. Specifically, we prove its joint continuity, a locally uniform sub-Gaussian upper bound of the form pt(x,y)≤C1t-1log(t-1)exp(-C2((|x-y|β∧1)/t)1β-1) for t∈(0,12] for each β>12(γ+2)2, and an on-diagonal lower bound of the form pt(x, x) ≥ C3t- 1(log (t- 1)) -η for t∈ (0 , tη(x) ] , with tη(x)∈(0,12]heavily dependent onx, for each η> 18 for Mγ-almost everyx. As applications, we deduce that the pointwise spectral dimension equals 2 Mγ-a.e. and that the global spectral dimension is also 2.