Parabolic Anderson Model in the Hyperbolic Space. Part II: Quenched Asymptotics
Xi Geng, Sheng Wang, Weijun Xu
TL;DR
This work proves sharp quenched long-time growth for the parabolic Anderson model on hyperbolic space with a regular Gaussian potential. By leveraging the Feynman–Kac representation and a non-Euclidean localisation mechanism, the authors obtain almost-sure asymptotics $u(t,x)=\exp(L^{*}t^{5/3}+o(t^{5/3}))$, with an explicit optimization-driven constant $L^{*}$ and optimisers $(\theta^{*},K^{*})$. The analysis combines precise peak-structure of the Gaussian field, hyperbolic Brownian motion geometry, and a two-scale clustering scheme to balance peak rewards against travel costs, yielding a unique, geometry-driven growth rate $t^{5/3}$ that contrasts with the Euclidean case. The results establish a sharp, geometry-sensitive analogue of PAM in a negatively curved setting and pave the way for further investigations into fluctuations and more singular potentials on manifolds.
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.