Superdiffusivity for brownian motion in a poissonian potential with long range correlation I: Lower bound on the volume exponent

Hubert Lacoin

Annales de l'I.H.P. Probabilités et statistiques (2012)

  • Volume: 48, Issue: 4, page 1010-1028
  • ISSN: 0246-0203

Abstract

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We study trajectories of d -dimensional Brownian Motion in Poissonian potential up to the hitting time of a distant hyper-plane. Our Poissonian potential V is constructed from a field of traps whose centers location is given by a Poisson Point Process and whose radii are IID distributed with a common distribution that has unbounded support; it has the particularity of having long-range correlation. We focus on the case where the law of the trap radii ν has power-law decay and prove that superdiffusivity hold under certain condition, and get a lower bound on the volume exponent. Results differ quite much with the one that have been obtained for the model with traps of bounded radii by Wühtrich (Ann. Probab.26 (1998) 1000–1015, Ann. Inst. Henri Poincaré Probab. Stat.34 (1998) 279–308): the superdiffusivity phenomenon is enhanced by the presence of correlation.

How to cite

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Lacoin, Hubert. "Superdiffusivity for brownian motion in a poissonian potential with long range correlation I: Lower bound on the volume exponent." Annales de l'I.H.P. Probabilités et statistiques 48.4 (2012): 1010-1028. <http://eudml.org/doc/272030>.

@article{Lacoin2012,
abstract = {We study trajectories of $d$-dimensional Brownian Motion in Poissonian potential up to the hitting time of a distant hyper-plane. Our Poissonian potential $V$ is constructed from a field of traps whose centers location is given by a Poisson Point Process and whose radii are IID distributed with a common distribution that has unbounded support; it has the particularity of having long-range correlation. We focus on the case where the law of the trap radii $\nu $ has power-law decay and prove that superdiffusivity hold under certain condition, and get a lower bound on the volume exponent. Results differ quite much with the one that have been obtained for the model with traps of bounded radii by Wühtrich (Ann. Probab.26 (1998) 1000–1015, Ann. Inst. Henri Poincaré Probab. Stat.34 (1998) 279–308): the superdiffusivity phenomenon is enhanced by the presence of correlation.},
author = {Lacoin, Hubert},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {streched polymer; quenched disorder; superdiffusivity; brownian motion; poissonian obstacles; correlation; stretched polymer; Brownian motion; random potential; Poissonian obstacles; hitting time; volume exponent},
language = {eng},
number = {4},
pages = {1010-1028},
publisher = {Gauthier-Villars},
title = {Superdiffusivity for brownian motion in a poissonian potential with long range correlation I: Lower bound on the volume exponent},
url = {http://eudml.org/doc/272030},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Lacoin, Hubert
TI - Superdiffusivity for brownian motion in a poissonian potential with long range correlation I: Lower bound on the volume exponent
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2012
PB - Gauthier-Villars
VL - 48
IS - 4
SP - 1010
EP - 1028
AB - We study trajectories of $d$-dimensional Brownian Motion in Poissonian potential up to the hitting time of a distant hyper-plane. Our Poissonian potential $V$ is constructed from a field of traps whose centers location is given by a Poisson Point Process and whose radii are IID distributed with a common distribution that has unbounded support; it has the particularity of having long-range correlation. We focus on the case where the law of the trap radii $\nu $ has power-law decay and prove that superdiffusivity hold under certain condition, and get a lower bound on the volume exponent. Results differ quite much with the one that have been obtained for the model with traps of bounded radii by Wühtrich (Ann. Probab.26 (1998) 1000–1015, Ann. Inst. Henri Poincaré Probab. Stat.34 (1998) 279–308): the superdiffusivity phenomenon is enhanced by the presence of correlation.
LA - eng
KW - streched polymer; quenched disorder; superdiffusivity; brownian motion; poissonian obstacles; correlation; stretched polymer; Brownian motion; random potential; Poissonian obstacles; hitting time; volume exponent
UR - http://eudml.org/doc/272030
ER -

References

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  1. [1] M. Balasz, J. Quastel and T. Seppäläinen. Fluctuation exponents for KPZ/stochastic Burgers equation. J. Amer. Math. Soc.24 (2011) 683–708. Zbl1227.60083MR2784327
  2. [2] E. Bolthausen. A note on diffusion of directed polymer in a random environment. Comm. Math. Phys.123 (1989) 529–534. Zbl0684.60013MR1006293
  3. [3] F. Comets and N. Yoshida. Directed polymers in a random environment are diffusive at weak disorder. Ann. Probab. 34 5 (2006) 1746–1770. Zbl1104.60061MR2271480
  4. [4] M. D. Donsker and S. R. S. Varadhan. Asymptotics for the Wiener sausage. Comm. Pure Appl. Math.28 (1975) 525–565. Zbl0333.60077MR397901
  5. [5] D. Ioffe and Y. Velenik. Crossing random walks and stretched polymers at weak disorder. Ann. Probab. To appear. Zbl1251.60074MR2952089
  6. [6] K. Johansson. Transversal fluctuation for increasing subsequences on the plane. Probab. Theory Related Fields116 (2000) 445–456. Zbl0960.60097MR1757595
  7. [7] M. Kardar, G. Parisi and Y. C. Zhang. Dynamic scaling of growing interface. Phys. Rev. Lett.56 (1986) 889–892. Zbl1101.82329
  8. [8] H. Lacoin. Influence of spatial correlation for directed polymers. Ann. Probab.39 (2011) 139–175. Zbl1208.82084MR2778799
  9. [9] H. Lacoin. Superdiffusivity for Brownian motion in a Poissonian Potential with long range correlation II: Upper bound on the volume exponent. Ann. Inst. Henri Poincaré Probab. Stat. To appear. Zbl1267.82147MR3052457
  10. [10] C. Licea, C. Newman and M. S. T. Piza. Superdiffusivity in first-passage percolation. Probab. Theory Related Fields106 (1996) 559–591. Zbl0870.60096MR1421992
  11. [11] 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. Zbl1041.60079MR2060455
  12. [12] L. A. Pastur. The behavior of certain Wiener integrals as t → ∞ and the density of states of Schrödinger equations with random potential. Teoret. Mat. Fiz.32 (1977) 88–95. Zbl0353.60053MR449356
  13. [13] M. Petermann. Superdiffusivity of directed polymers in random environment. Ph.D. thesis, Universität Zürich, 2000. 
  14. [14] T. Seppäläinen. Scaling for a one-dimensional directed polymer with constrained endpoints. Ann. Probab.40 (2012) 19–73. Zbl1254.60098
  15. [15] A. S. Sznitman. Shape theorem, Lyapounov exponents and large deviation for Brownian motion in Poissonian potential. Comm. Pure Appl. Math. 47 (1994) 1655–1688. Zbl0814.60022MR1303223
  16. [16] A. S. Sznitman. Distance fluctuations and Lyapounov exponents. Ann. Probab.24 (1996) 1507–1530. Zbl0871.60088MR1411504
  17. [17] A. S. Sznitman. Brownian Motion, Ostacles and Random Media. Springer, Berlin, 1998. Zbl0973.60003MR1717054
  18. [18] M. Wühtrich. Scaling identity for crossing Brownian motion in a Poissonian potential. Probab. Theory Related Fields112 (1998) 299–319. Zbl0938.60099MR1660910
  19. [19] M. Wühtrich. Superdiffusive behavior of two-dimensional Brownian motion in a Poissonian potential. Ann. Probab.26 (1998) 1000–1015. Zbl0935.60099MR1634412
  20. [20] M. Wühtrich. Fluctuation results for Brownian motion in a Poissonian potential. Ann. Inst. Henri Poincaré Probab. Stat.34 (1998) 279–308. Zbl0909.60073MR1625875

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