Correlation structure of intermittency in the parabolic Anderson model

Abstract

Consider the Cauchy problem ∂u(x, t)/∂t = ℋu(x, t) (x∈ℤd, t≥ 0) with initial condition u(x, 0) ≡ 1 and with ℋ the Anderson Hamiltonian ℋ = κΔ + ξ. Here Δ is the discrete Laplacian, κ∈ (0, ∞) is a diffusion constant, and ξ = {ξ(x): x∈ℤ d } is an i.i.d.random field taking values in ℝ. Gartner and Molchanov (1990) have shown that if the law of ξ(0) is nondegenerate, then the solution u is asymptotically intermittent. In the present paper we study the structure of the intermittent peaks for the special case where the law of ξ(0) is (in the vicinity of) the double exponential Prob(ξ(0) > s) = exp[−e s /θ] (s∈ℝ). Here θ∈ (0, ∞) is a parameter that can be thought of as measuring the degree of disorder in the ξ-field. Our main result is that, for fixed x, y∈ℤ d and t→∈, the correlation coefficient of u(x, t) and u(y, t) converges to ∥w ρ∥−2 l2Σz ∈ℤd w ρ(x+z)w ρ(y+z). In this expression, ρ = θ/κ while w ρ:ℤd→ℝ+ is given by w ρ = (v ρ)⊗ d with v ρ: ℤ→ℝ+ the unique centered ground state (i.e., the solution in l2(ℤ) with minimal l 2-norm) of the 1-dimensional nonlinear equation Δv + 2ρv log v = 0. The uniqueness of the ground state is actually proved only for large ρ, but is conjectured to hold for any ρ∈ (0, ∞). empty It turns out that if the right tail of the law of ξ(0) is thicker (or thinner) than the double exponential, then the correlation coefficient of u(x, t) and u(y, t) converges to δ x, y (resp.the constant function 1). Thus, the double exponential family is the critical class exhibiting a nondegenerate correlation structure.