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Correlation structure of intermittency in the parabolic Anderson model
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  • Published: April 1999

Correlation structure of intermittency in the parabolic Anderson model

  • J. Gärtner1 &
  • F. den Hollander2 

Probability Theory and Related Fields volume 114, pages 1–54 (1999)Cite this article

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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∈ℤd} is an i.i.d.random field taking values in ℝ. Gärtner 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∈ℤd 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.

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Authors and Affiliations

  1. Fachbereich Mathematik, Technische Universität Berlin, Strasse des 17. Juni 136, D-10623 Berlin, Germany, , , , , , DE

    J. Gärtner

  2. Mathematisch Instituut, Universiteit Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, The Netherlands, , , , , , NL

    F. den Hollander

Authors
  1. J. Gärtner
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  2. F. den Hollander
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Received: 5 March 1997 / Revised version: 21 September 1998

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Gärtner, J., den Hollander, F. Correlation structure of intermittency in the parabolic Anderson model. Probab Theory Relat Fields 114, 1–54 (1999). https://doi.org/10.1007/s004400050220

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  • Issue Date: April 1999

  • DOI: https://doi.org/10.1007/s004400050220

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  • Mathematics Subject Classification (1991): 60H25, 82C44(primary), 60F10, 60J15, 60J55 (secondary)
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