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

Neutrinos and Big-Bang Nucleosynthesis

Observations of clusters and super clusters of galaxies have indicated that the Universe is more dominated by baryons than ever estimated in the homogeneous cosmological model for primordial nucleosynthesis. Recent detections of possibly low deuterium abundance in Lyman-$\alpha$ clouds along the line of sight to high red-shift quasars have raised another potential difficulty that \he4 is overproduced in any cosmological models which satisfy the low deuterium abundance constraint. We show that the inhomogeneous cosmological model with degenerate electron-neutrino can resolve these two difficulties.Comment: 7 pages, latex, 3 figures. To appear in Nucl. Phys. A62