Geometric characterization of intermittency in the parabolic Anderson model

Abstract

We consider the parabolic Anderson problem ∂tu = ∆u + ξ(x)u on R+ × Z d with localized initial condition u(0, x) = δ0(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 ∆ + 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. Contents

Localisation in the Bouchaud–Anderson model

“…In particular, we show that this contribution is proportional to the principal eigenfunction of H restricted to E. This is similar in spirit to [12,Theorem 4.1], and it crucial to establishing complete localisation of the solution.…”

“…Methods and techniques. Our approach to proving localisation in the BAM is loosely based on existing techniques to prove localisation in the PAM (see, e.g., [1,12,26]), although the complex interaction between the potential field and the trapping landscape means that these techniques cannot be trivially adapted. Instead, the presence of the trapping landscape requires the development of existing techniques on two main fronts.…”

“…In Section 7 we complete the proof of Theorem 1.7 by showing that u 1 (t, z) is localised at the site Z t . The main idea is the same as in [12] and [26], namely that: (i) the solution u 1 (t, z) is asymptotically approximated by the principal eigenfunction of the operator ∆σ −1 + ξ restricted to the domain B t ; and (ii) the principal eigenfunction decays exponentially away from the site Z t . Underlying this reasoning is the fact that the domain B t has been constructed to ensure that λ (j) (Z t ) is the largest j-local principal eigenvalue in the domain.…”

“…It is similar in spirit to [12,Lemma 4.2] and [14, Lemma 2.18], however we will need an additional form of the bound to accommodate the impact of the trapping landscape (see the proof of Lemma 7.4). …”

“…This has been proven (in [20]) in the extremal 2 case of Pareto-like tail decay (g ξ (x) ∼ γ log x, for γ > d), and more recently (in [26] and [10]) in the case of Weibull-like tail decay (g ξ (x) ∼ x γ ). On the other hand, if log g ξ (x) ≫ x, then complete localisation is known not to hold (see [12]). What occurs in the interface regime of double-exponential tail decay (log g ξ (x) ∼ cx, for c > 0) is not currently well-understood.…”