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The Parabolic Anderson Model

In: Interacting Stochastic Systems

Author

Listed:
  • Jürgen Gärtner

    (Technische Universität Berlin, Institut für Mathematik, MA7-5)

  • Wolfgang König

    (Technische Universität Berlin, Institut für Mathematik, MA7-5)

Abstract

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.

Suggested Citation

  • Jürgen Gärtner & Wolfgang König, 2005. "The Parabolic Anderson Model," Springer Books, in: Jean-Dominique Deuschel & Andreas Greven (ed.), Interacting Stochastic Systems, pages 153-179, Springer.
    • Handle: RePEc:spr:sprchp:978-3-540-27110-9_8
    DOI: 10.1007/3-540-27110-4_8
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