Potential confinement property of the parabolic Anderson model

Gabriela Grüninger; Wolfgang König

Annales de l'I.H.P. Probabilités et statistiques (2009)

  • Volume: 45, Issue: 3, page 840-863
  • ISSN: 0246-0203

Abstract

top
We consider the parabolic Anderson model, the Cauchy problem for the heat equation with random potential in ℤd. We use i.i.d. potentials ξ:ℤd→ℝ in the third universality class, namely the class of almost bounded potentials, in the classification of van der Hofstad, König and Mörters [Commun. Math. Phys.267 (2006) 307–353]. This class consists of potentials whose logarithmic moment generating function is regularly varying with parameter γ=1, but do not belong to the class of so-called double-exponentially distributed potentials studied by Gärtner and Molchanov [Probab. Theory Related Fields111 (1998) 17–55]. In [Commun. Math. Phys.267 (2006) 307–353] the asymptotics of the expected total mass was identified in terms of a variational problem that is closely connected to the well-known logarithmic Sobolev inequality and whose solution, unique up to spatial shifts, is a perfect parabola. In the present paper we show that those potentials whose shape (after appropriate vertical shifting and spatial rescaling) is away from that parabola contribute only negligibly to the total mass. The topology used is the strong L1-topology on compacts for the exponentials of the potential. In the course of the proof, we show that any sequence of approximate minimisers of the above variational formula approaches some spatial shift of the minimiser, the parabola.

How to cite

top

Grüninger, Gabriela, and König, Wolfgang. "Potential confinement property of the parabolic Anderson model." Annales de l'I.H.P. Probabilités et statistiques 45.3 (2009): 840-863. <http://eudml.org/doc/78047>.

@article{Grüninger2009,
abstract = {We consider the parabolic Anderson model, the Cauchy problem for the heat equation with random potential in ℤd. We use i.i.d. potentials ξ:ℤd→ℝ in the third universality class, namely the class of almost bounded potentials, in the classification of van der Hofstad, König and Mörters [Commun. Math. Phys.267 (2006) 307–353]. This class consists of potentials whose logarithmic moment generating function is regularly varying with parameter γ=1, but do not belong to the class of so-called double-exponentially distributed potentials studied by Gärtner and Molchanov [Probab. Theory Related Fields111 (1998) 17–55]. In [Commun. Math. Phys.267 (2006) 307–353] the asymptotics of the expected total mass was identified in terms of a variational problem that is closely connected to the well-known logarithmic Sobolev inequality and whose solution, unique up to spatial shifts, is a perfect parabola. In the present paper we show that those potentials whose shape (after appropriate vertical shifting and spatial rescaling) is away from that parabola contribute only negligibly to the total mass. The topology used is the strong L1-topology on compacts for the exponentials of the potential. In the course of the proof, we show that any sequence of approximate minimisers of the above variational formula approaches some spatial shift of the minimiser, the parabola.},
author = {Grüninger, Gabriela, König, Wolfgang},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {parabolic Anderson problem; intermittency; logarithmic Sobolev inequality; potential shape; Feynman–Kac formula},
language = {eng},
number = {3},
pages = {840-863},
publisher = {Gauthier-Villars},
title = {Potential confinement property of the parabolic Anderson model},
url = {http://eudml.org/doc/78047},
volume = {45},
year = {2009},
}

TY - JOUR
AU - Grüninger, Gabriela
AU - König, Wolfgang
TI - Potential confinement property of the parabolic Anderson model
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2009
PB - Gauthier-Villars
VL - 45
IS - 3
SP - 840
EP - 863
AB - We consider the parabolic Anderson model, the Cauchy problem for the heat equation with random potential in ℤd. We use i.i.d. potentials ξ:ℤd→ℝ in the third universality class, namely the class of almost bounded potentials, in the classification of van der Hofstad, König and Mörters [Commun. Math. Phys.267 (2006) 307–353]. This class consists of potentials whose logarithmic moment generating function is regularly varying with parameter γ=1, but do not belong to the class of so-called double-exponentially distributed potentials studied by Gärtner and Molchanov [Probab. Theory Related Fields111 (1998) 17–55]. In [Commun. Math. Phys.267 (2006) 307–353] the asymptotics of the expected total mass was identified in terms of a variational problem that is closely connected to the well-known logarithmic Sobolev inequality and whose solution, unique up to spatial shifts, is a perfect parabola. In the present paper we show that those potentials whose shape (after appropriate vertical shifting and spatial rescaling) is away from that parabola contribute only negligibly to the total mass. The topology used is the strong L1-topology on compacts for the exponentials of the potential. In the course of the proof, we show that any sequence of approximate minimisers of the above variational formula approaches some spatial shift of the minimiser, the parabola.
LA - eng
KW - parabolic Anderson problem; intermittency; logarithmic Sobolev inequality; potential shape; Feynman–Kac formula
UR - http://eudml.org/doc/78047
ER -

References

top
  1. [1] N. H. Bingham, C. M. Goldie and J. L. Teugels. Regular Variation. Cambridge Univ. Press, 1987. Zbl0617.26001MR898871
  2. [2] M. Biskup and W. König. Long-time tails in the parabolic Anderson model with bounded potential. Ann. Probab. 29 (2001) 636–682. Zbl1018.60093MR1849173
  3. [3] E. Bolthausen. Localization of a two-dimensional random walk with an attractive path interaction. Ann. Probab. 22 (1994) 875–918. Zbl0819.60028MR1288136
  4. [4] E. A. Carlen. Superadditivity of Fisher’s information and logarithmic Sobolev inequality. J. Funct. Anal. 101 (1991) 194–211. Zbl0732.60020MR1132315
  5. [5] R. Carmona and S. A. Molchanov. Parabolic Anderson problem and intermittency. Mem. Amer. Math. Soc. 108 (1994) 1–125. Zbl0925.35074MR1185878
  6. [6] A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications, 2nd edition. Springer, New York, 1998. Zbl0896.60013MR1619036
  7. [7] J. Gärtner and F. den Hollander. Correlation structure of intermittency in the parabolic Anderson model. Probab. Theory Related Fields 114 (1999) 1–54. Zbl0951.60069MR1697138
  8. [8] J. Gärtner and W. König. The parabolic Anderson model. In Interacting Stochastic Systems 153–179. J.-D. Deuschel and A. Greven (Eds). Springer, Berlin, 2005. Zbl1111.82011MR2118574
  9. [9] J. Gärtner, W. König and S. Molchanov. Geometric characterization of intermittency in the parabolic Anderson model. Ann. Probab. 35 (2007) 439–499. Zbl1126.60091MR2308585
  10. [10] J. Gärtner and S. Molchanov. Parabolic problems for the Anderson model I. Intermittency and related topics. Commun. Math. Phys. 132 (1990) 613–655. Zbl0711.60055MR1069840
  11. [11] J. Gärtner and S. Molchanov. Parabolic problems for the Anderson model II. Second-order asymptotics and structure of high peaks. Probab. Theory Related Fields 111 (1998) 17–55. Zbl0909.60040MR1626766
  12. [12] R. van der Hofstad, W. König and P. Mörters. The universality classes in the parabolic Anderson model. Commun. Math. Phys. 267 (2006) 307–353. Zbl1115.82030
  13. [13] E. H. Lieb and M. Loss. Analysis, 2nd edition. AMS Graduate Studies 14. Amer. Math. Soc., Providence, RI, 2001. Zbl0966.26002MR1817225
  14. [14] S. Molchanov. Lectures on random media. In Lectures on Probability Theory. Ecole d’Eté de Probabilités de Saint-Flour XXII-1992242–411. D. Bakry, R. D. Gill and S. Molchanov (Eds). Lecture Notes in Math. 1581. Springer, Berlin, 1994. Zbl0814.60093MR1307415
  15. [15] T. Povel. Confinement of Brownian motion among Poissonian obstacles in Rd, d≥3. Probab. Theory Related Fields 114 (1999) 177–205. Zbl0943.60082MR1701519
  16. [16] A.-S. Sznitman. On the confinement property of two-dimensional Brownian motion among Poissonian obstacles. Comm. Pure Appl. Math. 44 (1991) 1137–1170. Zbl0753.60075MR1127055

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.