Superdiffusivity for brownian motion in a poissonian potential with long range correlation II: Upper bound on the volume exponent
Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 4, pp. 1029-1048.

Cet article est la seconde partie d’une étude sur les trajectoires Brownienne dans un champs de pièges mous dont le rayon est aléatoire et a une distribution non-bornée. Nous montrons que l’exposant de volume (qui est l’exposant associé aux fluctuations transversales des trajectoires) ξ est strictement inférieur à 1 et nous donnons une borne supérieure explicite qui dépend des paramètres du problème, et ceci aussi bien pour le modèle dans la configuration point-à-point que pour celui dans la configuration point à plan. Dans certains cas particulier, cette borne supérieure coïncide avec la borne inférieure démontrée dans la première partie de cette étude, ce qui nous permets d’identifier la valeur de l’exposant de volume.

This paper continues a study on trajectories of Brownian Motion in a field of soft trap whose radius distribution is unbounded. We show here that for both point-to-point and point-to-plane model the volume exponent (the exponent associated to transversal fluctuation of the trajectories) ξ is strictly less than 1 and give an explicit upper bound that depends on the parameters of the problem. In some specific cases, this upper bound matches the lower bound proved in the first part of this work and we get the exact value of the volume exponent.

DOI : 10.1214/11-AIHP457
Classification : 82D60, 60K37, 82B44
Mots clés : stretched polymer, quenched disorder, superdiffusivity, brownian motion, poissonian obstacles, correlation
@article{AIHPB_2012__48_4_1029_0,
     author = {Lacoin, Hubert},
     title = {Superdiffusivity for brownian motion in a poissonian potential with long range correlation {II:} {Upper} bound on the volume exponent},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {1029--1048},
     publisher = {Gauthier-Villars},
     volume = {48},
     number = {4},
     year = {2012},
     doi = {10.1214/11-AIHP457},
     mrnumber = {3052457},
     zbl = {1267.82147},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/11-AIHP457/}
}
TY  - JOUR
AU  - Lacoin, Hubert
TI  - Superdiffusivity for brownian motion in a poissonian potential with long range correlation II: Upper bound on the volume exponent
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2012
SP  - 1029
EP  - 1048
VL  - 48
IS  - 4
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/11-AIHP457/
DO  - 10.1214/11-AIHP457
LA  - en
ID  - AIHPB_2012__48_4_1029_0
ER  - 
%0 Journal Article
%A Lacoin, Hubert
%T Superdiffusivity for brownian motion in a poissonian potential with long range correlation II: Upper bound on the volume exponent
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2012
%P 1029-1048
%V 48
%N 4
%I Gauthier-Villars
%U http://www.numdam.org/articles/10.1214/11-AIHP457/
%R 10.1214/11-AIHP457
%G en
%F AIHPB_2012__48_4_1029_0
Lacoin, Hubert. Superdiffusivity for brownian motion in a poissonian potential with long range correlation II: Upper bound on the volume exponent. Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 4, pp. 1029-1048. doi : 10.1214/11-AIHP457. http://www.numdam.org/articles/10.1214/11-AIHP457/

[1] M. Balasz, J. Quastel and T. Seppäläinen. Fluctuation exponent of the KPZ/stochastic Burgers equation. J. Amer. Math. Soc. 24 (2011) 683-708. | MR | Zbl

[2] K. Johansson. Transversal fluctuation for increasing subsequences on the plane. Probab. Theory Related Fields 116 (2000) 445-456. | MR | Zbl

[3] H. Kesten. On the speed of convergence in first-passage percolation. Ann. Appl. Probab. 3 (1993) 296-338. | MR | Zbl

[4] H. Lacoin. Influence of spatial correlation for directed polymers. Ann. Probab. 39 (2011) 139-175. | MR | Zbl

[5] H. Lacoin. Superdiffusivity for Brownian motion in a Poissonian potential with long range correlation I: Lower bound on the volume exponent. Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012) 1010-1028. | Numdam | MR | Zbl

[6] O. Méjane. Upper bound of a volume exponent for directed polymers in a random environment. Ann. Inst. Henri Poincaré Probab. Stat. 40 (2004) 299-308. | Numdam | MR | Zbl

[7] T. Seppäläinen. Scaling for a one-dimensional directed polymer with boundary conditions. Ann. Probab. 40 (2012) 19-73. | MR | Zbl

[8] A. S. Sznitman. Shape theorem, Lyapounov exponents and large deviation for Brownian motion in Poissonian potential. Comm. Pure Appl. Math. 47 (1994) 1655-1688. | MR | Zbl

[9] A. S. Sznitman. Distance fluctuations and Lyapounov exponents. Ann. Probab. 24 (1996) 1507-1530. | MR | Zbl

[10] A. S. Sznitman. Brownian Motion, Obstacles and Random Media. Springer Monographs in Mathematics. Springer, Berlin, 1998. | MR | Zbl

[11] M. Wütrich. Superdiffusive behavior of two-dimensional Brownian motion in a Poissonian potential. Ann. Probab. 26 (1998) 1000-1015. | MR | Zbl

[12] M. Wütrich. Fluctuation results for Brownian motion in a Poissonian potential. Ann. Inst. Henri Poincaré Probab. Stat. 34 (1998) 279-308. | Numdam | MR | Zbl

Cité par Sources :