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

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 measure Mγ, 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 on x, for each η>18 for Mγ -almost every x. As applications, we deduce that the pointwise spectral dimension equals 2 Mγ-a.e. and that the global spectral dimension is also 2.