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 $2$.Comment: 36 page

The Laplacian on some self-conformal fractals and Weyl's asymptotics for its eigenvalues: A survey of the ergodic-theoretic aspects (Research on the Theory of Random Dynamical Systems and Fractal Geometry)

This short survey is aimed at sketching the ergodic-theoretic aspects of the author's recent studies on Weyl's eigenvalue asymptotics for a "geometrically canonical" Laplacian defined by the author on some self-conformal circle packing fractals including the classical Apollonian gasket. The main result being surveyed is obtained by applying Kesten's renewal theorem [Ann. Probab. 2 (1974), 355- 386, Theorem 2] for functionals of Markov chains on general state spaces and provides an alternative probabilistic proof of the result by Oh and Shah [Invent. Math. 187 (2012), 1-35, Corollary 1.8] on the asymptotic distribution of the circles in the Apollonian gasket

A bridge between elliptic and parabolic Harnack inequalities

The notion of conformal walk dimension serves as a bridge between elliptic and parabolic Harnack inequalities. The importance of this notion is due to the fact that the finiteness of the conformal walk dimension characterizes the elliptic Harnack inequality. Roughly speaking, the conformal walk dimension is the infimum of all possible values of the walk dimension that can be attained by a time-change of the process and by a quasisymmetric change of the metric. We show that the conformal walk dimension of any space satisfying the elliptic Harnack inequality is two. We also provide examples that show that the infimum in the definition of conformal walk dimension may or may not be attained.Comment: 100 pages; preliminary version; comments are welcom

Non-Regularly Varying and Non-Periodic Oscillation of the On-Diagonal Heat Kernels on Self-Similar Fractals

Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II: Fractals in Applied Mathematic