The parabolic Anderson model

2004/03/04 by Juergen Gaertner, Wolfgang Koenig, Gaertner, Juergen +1 · 3 citations
Mathematics · #60H25 #82C44 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60H25 #msc:82C44

paper · pdf · doi:10.48550/arxiv.math/0403091

21 pages

arxiv created 2004/03/04 · arxiv updated 2009/12/01

Abstract

This is a survey on the intermittent behavior of the parabolic Anderson model, which is the Cauchy problem for the heat equation with random potential on the lattice \Z^d. We first introduce the model and give heuristic explanations of the long-time behavior of the solution, both in the annealed and the quenched setting for time-independent potentials. We thereby consider examples of potentials studied in the literature. In the particularly important case of an i.i.d. potential with double-exponential tails we formulate the asymptotic results in detail. Furthermore, we explain that, under mild regularity assumptions, there are only four different universality classes of asymptotic behaviors. Finally, we study the moment Lyapunov exponents for space-time homogeneous catalytic potentials generated by a Poisson field of random walks.

Cited by

• Renormalisation of singular SPDEs with Correlated Coefficients
• On the Spectral Geometry and Small Time Mass of Anderson Models on Planar Domains
• Parabolic Anderson Model in the Hyperbolic Space. Part II: Quenched Asymptotics

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