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

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 $$u(t,x)\sim e^{L^{\ast }{t}^{5\mathrm{/}3}+o(t^{5\mathrm{/}3})}$$ as $t\to +\mathrm{\infty }$. Both the power $t^{5\mathrm{/}3}$ on the exponential and the exact value of $L^{\ast }$ 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.