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

We establish the second-order moment asymptotics for a parabolic Anderson model ${\mathrm{\partial }}_{t}u=(\mathrm{\Delta }+\xi )u$ 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.