Liouville Brownian motion and quantum cones in dimension $d>2$

Abstract

For $d>2$ and $\gamma \in (0,\sqrt{2d})$, we study the Liouville Brownian motion associated with the whole-space log-correlated Gaussian field in ${\mathbb{R}}^d$. We compute its spectral dimension, i.e., the short-time asymptotics of the heat kernel along the diagonal, which, in contrast to the two-dimensional case, depends on both $\gamma$ and on the thickness of the starting point. Furthermore, for even dimensions $d>2$, we show that the spherical average process of the whole-space log-correlated Gaussian field in ${\mathbb{R}}^d$ can be identified with the integral of a stationary Gaussian Markov process of order $(d-2)/2$. Exploiting this representation, we construct the higher-dimensional analogue of the $\beta$-quantum cone for $\beta \in (-\infty ,Q)$, with $Q=d/\gamma +\gamma /2$. Lastly, for $\alpha =Q-\sqrt{Q^2-4}$, we prove that the law of the $d$-dimensional $\alpha$-quantum cone is invariant under shifts along the trajectories of the associated Liouville Brownian motion.

51 pages, 2 figures