Parabolic Anderson Model in the Hyperbolic Space. Part I: Annealed Asymptotics

TL;DR

This paper analyzes the second-order moment asymptotics of the parabolic Anderson model in hyperbolic space with Gaussian potential, revealing that growth and fluctuations mirror Euclidean behavior due to a curvature dilation effect.

Contribution

It establishes second-order moment asymptotics for the model in hyperbolic space, showing Euclidean-like intermittency and identifying the fluctuation exponent via a Euclidean-induced variational problem.

Findings

Growth and fluctuation asymptotics match Euclidean case

Solution exhibits Euclidean-like intermittency

Fluctuation exponent determined by Euclidean Laplacian variational problem

Abstract

We establish the second-order moment asymptotics for a parabolic Anderson model $\partial_{t}u=(\Delta+\xi)u$ in the hyperbolic space with a regular, stationary Gaussian potential $\xi$. It turns out that the growth and fluctuation asymptotics both are identical to the Euclidean situation. As a result, the solution exhibits the same moment intermittency property as in the Euclidean case. An interesting point here is that the fluctuation exponent is determined by a variational problem induced by the Euclidean (rather than hyperbolic) Laplacian. Heuristically, this is due to a curvature dilation effect: the geometry becomes asymptotically flat after suitable renormalisation in the derivation of the second-order asymptotics.