Advertisement

The Parabolic Anderson Model with Acceleration and Deceleration

  • Wolfgang König
  • Sylvia Schmidt
Conference paper
Part of the Springer Proceedings in Mathematics book series (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.

Keywords

Local Time Potential Distribution Variational Formula Universality Class Large Deviation Principle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Becker, M., König, W.: Self-intersection local times of random walks: Exponential moments in subcritical dimensions, Probab. Theory Relat. Fields (to appear)Google Scholar
  2. 2.
    Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge (1987)zbMATHGoogle Scholar
  3. 3.
    Biskup, M., König, W.: Long-time tails in the parabolic Anderson model with bounded potential. Ann. Probab. 29(2), 636–682 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Braess, D.: Finite elements. Theory, fast solvers and applications in elasticity theory (Finite Elemente. Theorie, schnelle Löser und Anwendungen in der Elastizitätstheorie), 4th revised and extended edn. (German). Springer, Berlin (2007)Google Scholar
  5. 5.
    Brydges, D., van der Hofstad, R., König, W.: Joint density for the local times of continuous-time Markov chains. Ann. Probab. 35(4), 1307–1332 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Carmona, R.A., Molchanov, S.A.: Parabolic Anderson problem and intermittency. Mem. Am. Math. Soc. 518, 125 (1994)MathSciNetGoogle Scholar
  7. 7.
    Coudière, Y., Gallouét, T., Herbin, R.: Discrete Sobolev inequalities and L perror estimates for finite volume solutions of convection diffusion equations. M2AN, Math. Model. Numer. Anal. 35(4), 767–778 (2001)Google Scholar
  8. 8.
    Donsker, M.D., Varadhan, S.R.S.: Asymptotics for the Wiener sausage. Commun. Pure Appl. Math. 28, 525–565 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Gantert, N., König, W., Shi, Z.: Annealed deviations of random walk in random scenery. Ann. Inst. Henri Poincaré, Probab. Stat. 43(1), 47–76 (2007)Google Scholar
  10. 10.
    Gärtner, J., den Hollander, F.: Correlation structure of intermittency in the parabolic Anderson model. Probab. Theory Relat. Fields 114(1), 1–54 (1999)zbMATHCrossRefGoogle Scholar
  11. 11.
    Gärtner, J., König, W.: The parabolic Anderson model. In: Deuschel, J.-D., et al. (ed.) Interacting Stochastic Systems, pp. 153–179. Springer, Berlin (2005)CrossRefGoogle Scholar
  12. 12.
    Gärtner, J., Molchanov, S.A.: Parabolic problems for the Anderson model. I: Intermittency and related topics. Commun. Math. Phys. 132(3), 613–655 (1990)zbMATHCrossRefGoogle Scholar
  13. 13.
    Gärtner, J., Molchanov, S.A.: Parabolic problems for the Anderson model. II: Second-order asymptotics and structure of high peaks. Probab. Theory Relat. Fields 111(1), 17–55 (1998)zbMATHCrossRefGoogle Scholar
  14. 14.
    Grüninger, G., König, W.: Potential confinement property of the parabolic Anderson model. Ann. Inst. Henri Poincaré, Probab. Stat. 45(3), 840–863 (2009)Google Scholar
  15. 15.
    van der Hofstad, R., König, W., Mörters, P.: The universality classes in the parabolic Anderson model. Commun. Math. Phys. 267(2), 307–353 (2006)zbMATHCrossRefGoogle Scholar
  16. 16.
    Schmidt, B.: On a semilinear variational problem. ESAIM Control Optim. Calc. Var. 17, 86–101 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Schmidt, S.: Das parabolische Anderson-Modell mit Be- und Entschleunigung (German), PhD thesis, University of Leipzig (2010) and SVH Saarbrücken (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Weierstraß-Institut BerlinBerlinGermany
  2. 2.Institut für MathematikTechnische Universität BerlinBerlinGermany
  3. 3.Kompetenzzentrum für klinische StudienUniversität BremenBremenGermany

Personalised recommendations