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The Parabolic Anderson Model

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Summary

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 ℤd. 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.

Partially supported by the German Science Foundation, Schwerpunkt project SPP 1033

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Authors and Affiliations

  1. Institut für Mathematik, MA7-5, Technische Universität Berlin, Straße des 17. Juni 136, D-10623, Berlin, Germany

    Jürgen Gärtner & Wolfgang König

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  1. Jürgen Gärtner
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  2. Wolfgang König
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Editors and Affiliations

  1. Fakultät II, Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Jean-Dominique Deuschel

  2. Fachbereich Mathematik und Physik Mathematisches Institut, Universität Erlangen, Bismarckstr. 1 1/2, 91054, Erlangen, Germany

    Andreas Greven

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Gärtner, J., König, W. (2005). The Parabolic Anderson Model. In: Deuschel, JD., Greven, A. (eds) Interacting Stochastic Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-27110-4_8

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