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 $$p_t(x,y)\le C_{1} t^{-1} \log (t^{-1}) \exp \bigl (-C_{2}((|x-y|^{\beta }\wedge 1)/t)^{\frac{1}{\beta -1}}\bigr )$$ for $$t\in \bigl (0,\frac{1}{2}\bigr ]$$ for each $$\beta >\frac{1}{2}(\gamma +2)^2$$ , and an on-diagonal lower bound of the form $$p_{t}(x,x)\ge C_{3}t^{-1}\bigl (\log (t^{-1})\bigr )^{-\eta }$$ for $$t\in (0,t_{\eta }(x)]$$ , with $$t_{\eta }(x)\in \bigl (0,\frac{1}{2}\bigr ]$$ 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. [ABSTRACT FROM AUTHOR]