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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 $M_γ$, formally written as $M_γ(dz)=e^{γX(z)-{γ^2} \mathbb{E}[X(z)^2]/2}\, dz$, $γ\in(0,2)$, for a (massive) Gaussian free field $X$. It is an $M_γ$-symmetric diffusion defined as the time change of the two-dimensional Brownian motion by the positive continuous additive functional with Revuz measure $M_γ$. 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 $p_t(x,y)\leq C_{1} t^{-1} \log(t^{-1}) \exp\bigl(-C_{2}((|x-y|^β\wedge 1)/t)^{\frac{1}{β-1}}\bigr)$ for $t\in(0,\frac{1}{2}]$ for each $β>\frac{1}{2}(γ+2)^2$, and an on-diagonal lower bound of the form $p_{t}(x,x)\geq C_{3}t^{-1}\bigl(\log(t^{-1})\bigr)^{-η}$ for $t\in(0,t_η(x)]$, with $t_η(x)\in(0,\frac{1}{2}]$ heavily dependent on $x$, for each $η>18$ for $M_γ$-almost every $x$. As applications, we deduce that the pointwise spectral dimension equals $2$ $M_γ$-a.e.\ and that the global spectral dimension is also $2$.