Correlation structure of intermittency in the parabolic Anderson model

Abstract

Consider the Cauchy problem @u(x t)=@t = Hu(x t) ( x 2 Z Z d t 0) with initial condition u(x 0) 1 and with H the Anderson Hamiltonian H = + . Here is the discrete Laplacian, 2 (0 1) is a di usion constant, and = f (x): x 2 Z Z d g is an i.i.d. random eld taking values in IR. G artner and Molchanov (1990) have shown that if the law o f (0) is nondegenerate, then the solution u is asymptotically intermittent. This means that lim t!1 hu 2 (0 t )i=hu(0 t )i 2 = 1, where h i denotes expectation w.r.t. , and similarly for the higher moments. Qualitatively their result says that, as t increases, the random eld fu(x t): x 2 ZZ d g develops sparsely distributed high peaks, which g i v e the dominant c o n tribution to the moments as they become sparser and higher.In the present paper we study the structure of the intermittent peaks for the special case where the law o f (0) is (in the vicinity of) the double exponential Prob( (0) > s ) = exp ;e s= ] ( s 2 IR). Here 2 (0 1) is a parameter that can be thought of as measuring the degree of disorder in the -eld. Our main result is that, for xed x y 2 Z Z d and t ! 1 , the correlation coe cient o f u(x t) and u(y t) c o n verges to kw k ;2 2 (Z Z) with minimal l 2 -norm). Qualitatively our result says that the high peaks of u have a shape that is a multiple of w relative t o t h e c e n ter of the peak. It will turn out that if the right tail of the law o f (0) is thicker (or thinner) than the double exponential, then the correlation coe cient o f u(x t) and u(y t) c o n verges to x y (resp. the constant function 1). Thus, the double exponential family is the critical class exhibiting a nondegenerate correlation structure.1991 Mathematics Subject Classi cation: 60H25, 82C44 (primary), 60F10, 60J15, 60J55 (secondary).

The Universality Classes in the Parabolic Anderson Model

“…the discrete variant of χ(ρ) in (1.23), which was studied in Gärtner and den Hollander [GH99]. In Proposition 3 and the subsequent remark they show that…”

“…It has a solution, which is unique for sufficiently large values of ρ, and heuristically this minimizer represents the shape of the solution. As noted in [GH99], for any family of minimizers g ρ , as ρ ↑ ∞, g ρ converges to δ 0 , which links to the single-peak case. Furthermore, as ρ ↓ 0, the minimisers g ρ are asymptotically given by…”

Almost sure asymptotics for the continuous parabolic Anderson model

“…The field ξ under investigation there (the so-called double-exponential distribution) does not lead to a spatial scaling (in contrast to the present paper); the main contribution comes from peaks of fixed finite size. The correlation structure of the solution has been studied by Gärtner and den Hollander [4].…”

Mass concentration and aging in the parabolic Anderson model with doubly-exponential tails

“…In particular, M * ρ = ∅ and, by Lemma 3.1 therein, all V ∈ M * ρ satisfy L(V) = 1. On the other hand, by (3.21) in [GKM07] together with Theorem 2 and Proposition 3 of [GH99] (see also (5.44) therein),…”

“…We will now combine the previous two lemmas with results from [BK16], [GH99] and [GKM07] to obtain upper and lower bounds around a L for the potential in relevant islands. …”