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

Abstract

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 $p_t(x,y)\le C_1{t}^{-1}\log (t^{-1})\exp (-C_2((|x-y{|}^{\beta }\wedge 1)/t{)}^{\frac{1}{\beta -1}})$ 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 $p_t(x,x)\ge C_3{t}^{-1}(\log (t^{-1}){)}^{-\eta }$ 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 $2$.

36 pages

References in corpus (2)

• KPZ formula derived from Liouville heat kernel
• Liouville heat kernel: regularity and bounds

Cited by in corpus (4)

• Liouville heat kernel: regularity and bounds
• Elliptic regularity results: n-regularized Liouville Brownian motion and non-symmetric diffusions associated with degenerate forms
• Heat kernels on 2d Liouville quantum gravity: a numerical study
• Localized upper bounds of heat kernels for diffusions via a multiple Dynkin-Hunt formula