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 $M_\gamma$, 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 $p_t(x,y)\leq C_{1} t^{-1} \log(t^{-1}) \exp\bigl(-C_{2}((|x-y|^{\beta}\wedge 1)/t)^{\frac{1}{\beta -1}}\bigr)$ for $t\in(0,\frac{1}{2}]$ for each $\beta>\frac{1}{2}(\gamma+2)^2$, 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 $t\in(0,t_{\eta}(x)]$, with $t_{\eta}(x)\in(0,\frac{1}{2}]$ heavily dependent on $x$, for each $\eta>18$ for $M_{\gamma}$-almost every $x$. As applications, we deduce that the pointwise spectral dimension equals $2$ $M_\gamma$-a.e.\ and that the global spectral dimension is also $2$.