Parabolic Anderson Model in the Hyperbolic Space. Part II: Quenched Asymptotics

TL;DR

This paper determines the precise long-term growth rate of solutions to the parabolic Anderson model in hyperbolic space with Gaussian potential, revealing differences from Euclidean cases and employing advanced localization techniques.

Contribution

It provides the exact quenched asymptotic growth rate of PAM solutions in hyperbolic space, highlighting novel non-Euclidean localization mechanisms and explicit optimization methods.

Findings

Solution grows as e^{L^{*}t^{5/3}+o(t^{5/3})} almost surely.

Growth rate and exponent differ from Euclidean case, with explicit L^*.

Introduces new localization techniques specific to hyperbolic geometry.

Abstract

We establish the exact quenched asymptotic growth of the solution to the parabolic Anderson model (PAM) in the hyperbolic space with a regular, stationary, time-independent Gaussian potential. More precisely, we show that with probability one, the solution $u$ to PAM with constant initial data has pointwise growth asymptotics \[ u(t,x)\sim e^{L^{*}t^{5/3}+o(t^{5/3})} \] as $t \rightarrow +\infty$. Both the power $t^{5/3}$ on the exponential and the exact value of $L^*$ are different from their counterparts in the Euclidean situation. They are determined through an explicit optimisation procedure. Our proof relies on certain fine localisation techniques, which also reveals a stronger non-Euclidean localisation mechanism.