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

Sebastian Andres, Naotaka Kajino

Published 2014-07-11, updated 2015-01-30Version 2

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 $M_\gamma$, formally written as $M_\gamma(dz)=e^{\gamma X(z)-{\gamma^2} \mathbb{E}[X(z)^2]/2} dz$, $\gamma\in(0,2)$, for a (massive) Gaussian free field $X$. It is an $M_\gamma$-symmetric diffusion defined as the time change of the two-dimensional Brownian motion by the positive continuous additive functional with Revuz measure $M_\gamma$. In this paper we provide a detailed analysis of the heat kernel $p_t(x,y)$ of the LBM. Specifically, we prove its joint continuity, a locally uniform sub-Gaussian upper bound of the form $p_t(x,y)\leq C_{1} t^{-1} \log(t^{-1}) \exp\bigl(-C_{2}(|x-y|^{\beta}/t)^{\frac{1}{\beta-1}}\bigr)$ for small $t$ for each $\beta>(\gamma+2)^2/2$ and an on-diagonal lower bound of the form $p_{t}(x,x)\geq C_{3}t^{-1}\bigl(\log(t^{-1})\bigr)^{-\eta}$ for small $t$ for $M_{\gamma}$-a.e.\ $x$ with some concrete constant $\eta>0$. As applications, we also show that the pointwise spectral dimension equals $2$ $M_\gamma$-a.e.\ and that the global spectral dimension is $2$ as well.