WIAS Preprint No. 1552, (2010)

The parabolic Anderson model with acceleration and deceleration


  • König, Wolfgang
    ORCID: 0000-0002-7673-4364
  • Schmidt, Sylvia

2010 Mathematics Subject Classification

  • 35K15 82B44 60F10, 60K37


  • parabolic Anderson model, moment asymptotics, variational formulas, accelerated and decelerated diffusion, large deviations, random walk in random scenery




We describe the large-time moment asymptotics for the parabolic Anderson model where the speed of the diffusion is coupled with time, inducing an acceleration or deceleration. We find a lower critical scale, below which the mass flow gets stuck. On this scale, a new interesting variational problem arises in the description of the asymptotics. Furthermore, we find an upper critical scale above which the potential enters the asymptotics only via some average, but not via its extreme values. We make out altogether five phases, three of which can be described by results that are qualitatively similar to those from the constant-speed parabolic Anderson model in earlier work by various authors. Our proofs consist of adaptations and refinements of their methods, as well as a variational convergence method borrowed from finite elements theory.

Appeared in

  • Probability in Complex Physical Systems, in Honour of Erwin Bolthausen and Jürgen Gärtner, J.-D. Deuschel, B. Gentz, W. König, M. von Renesse, M. Scheutzow, eds., vol. 11 of Springer Proceedings in Mathematics, Springer, Berlin Heidelberg, 2012, pp. 225--245

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