Mass concentration and aging in the parabolic Anderson model with doubly-exponential tails release_kqjj72w3mrgijp2ab7sxcxin3e

by Marek Biskup, Wolfgang Koenig, Renato Soares dos Santos

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(2016)

Abstract

We study the solutions u=u(x,t) to the Cauchy problem on Z^d×(0,∞) for the parabolic equation ∂_t u=Δ u+ξ u with initial data u(x,0)=1_{0}(x). Here Δ is the discrete Laplacian on Z^d and ξ=(ξ(z))_z∈ Z^d is an i.i.d.random field with doubly-exponential upper tails. We prove that, for large t and with large probability, a majority of the total mass U(t):=∑_x u(x,t) of the solution resides in a bounded neighborhood of a site Z_t that achieves an optimal compromise between the local Dirichlet eigenvalue of the Anderson Hamiltonian Δ+ξ and the distance to the origin. The processes t Z_t and t 1t U(t) are shown to converge in distribution under suitable scaling of space and time. Aging results for Z_t, as well as for the solution to the parabolic problem, are also established. The proof uses the characterization of eigenvalue order statistics for Δ+ξ in large sets recently proved by the first two authors.
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Date   2016-09-04
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arXiv  1609.00989v1
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