Chapter

Interacting Stochastic Systems

pp 153-179

The Parabolic Anderson Model

  • Jürgen GärtnerAffiliated withInstitut für Mathematik, MA7-5, Technische Universität Berlin
  • , Wolfgang KönigAffiliated withInstitut für Mathematik, MA7-5, Technische Universität Berlin

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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.

Key words

Parabolic Anderson problem heat equation with random potential intermittency Feynman-Kac formula random environment