Geometric characterization of intermittency in the parabolic Anderson model
Abstract
We consider the parabolic Anderson problem
∂
t
u=Δu+ξ(x)u
on
R
+
×
Z
d
with localized initial condition
u(0,x)=
δ
0
(x)
and random i.i.d. potential
ξ
. Under the assumption that the distribution of
ξ(0)
has a double-exponential, or slightly heavier, tail, we prove the following geometric characterization of intermittency: with probability one, as
t→∞
, the overwhelming contribution to the total mass
∑
x
u(t,x)
comes from a slowly increasing number of ``islands'' which are located far from each other. These ``islands'' are local regions of those high exceedances of the field
ξ
in a box of side length
2t
log
2
t
for which the (local) principal Dirichlet eigenvalue of the random operator
Δ+ξ
is close to the top of the spectrum in the box. We also prove that the shape of
ξ
in these regions is nonrandom and that
u(t,⋅)
is close to the corresponding positive eigenfunction. This is the geometric picture suggested by localization theory for the Anderson Hamiltonian.