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The Parabolic Anderson Model with Acceleration and Deceleration

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Probability in Complex Physical Systems

Part of the book series: Springer Proceedings in Mathematics ((PROM,volume 11))

Abstract

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.

MSC 2000:  35K15, 82B44, 60F10, 60K37.

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Author information

Authors and Affiliations

  1. Weierstraß-Institut Berlin, Mohrenstr. 39, 10117, Berlin, Germany

    Wolfgang König

  2. Institut für Mathematik, Technische Universität Berlin, Str. des 17. Juni 136, 10623, Berlin, Germany

    Wolfgang König

  3. Kompetenzzentrum für klinische Studien, Universität Bremen, Achterstr. 30, 28359, Bremen, Germany

    Sylvia Schmidt

Authors
  1. Wolfgang König
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  2. Sylvia Schmidt
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Corresponding author

Correspondence to Wolfgang König .

Editor information

Editors and Affiliations

  1. Institute for Mathematics, TU Berlin, Straße des 17. Juni 136, Berlin, 10623, Berlin, Germany

    Jean-Dominique Deuschel

  2. Faculty of Mathematics, University of Bielefeld, P.O. Box 10 01 31, Bielefeld, 33501, Germany

    Barbara Gentz

  3. for Applied Analysis and Stochastics, Weierstrass Institute Berlin, Mohrenstr. 39, Berlin, 10117, Germany

    Wolfgang König

  4. Institute for Mathematics, TU Berlin, Straße des 17. Juni 136, Berlin, 10623, Germany

    Max von Renesse

  5. Institute for Mathematics, TU Berlin, Straße des 17. Juni 136, Berlin, 10623, Germany

    Michael Scheutzow

  6. Institute for Mathematical, Methods in Economics, Vienna University of Technology, Wiedner Hauptstraße 8-10/105-1, Vienna, 1040, Austria

    Uwe Schmock

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König, W., Schmidt, S. (2012). The Parabolic Anderson Model with Acceleration and Deceleration. In: Deuschel, JD., Gentz, B., König, W., von Renesse, M., Scheutzow, M., Schmock, U. (eds) Probability in Complex Physical Systems. Springer Proceedings in Mathematics, vol 11. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23811-6_9

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