no. 3
Einstein relation for biased random walk on Galton-Watson trees
Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 3, pp. 698-721.

Nous prouvons la relation d'Einstein pour certaines marches aléatoires biaisées sur des arbres de Galton-Watson. Cette formule relie la dérivée de la vitesse à la diffusivité à l'équilibre. Ce travail fournit le premier exemple de preuve de la relation d'Einstein pour une dynamique dans un milieu aléatoire qui comporte des pièges arbitrairement lents.

We prove the Einstein relation, relating the velocity under a small perturbation to the diffusivity in equilibrium, for certain biased random walks on Galton-Watson trees. This provides the first example where the Einstein relation is proved for motion in random media with arbitrarily slow traps.

DOI : 10.1214/12-AIHP486
Classification : 60K37, 60J80, 82C44
Mots clés : Galton-Watson tree, Einstein relation, spine representation 
@article{AIHPB_2013__49_3_698_0,
     author = {Ben Arous, Gerard and Hu, Yueyun and Olla, Stefano and Zeitouni, Ofer},
     title = {Einstein relation for biased random walk on {Galton-Watson} trees},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {698--721},
     publisher = {Gauthier-Villars},
     volume = {49},
     number = {3},
     year = {2013},
     doi = {10.1214/12-AIHP486},
     mrnumber = {3112431},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/12-AIHP486/}
}
TY  - JOUR
AU  - Ben Arous, Gerard
AU  - Hu, Yueyun
AU  - Olla, Stefano
AU  - Zeitouni, Ofer
TI  - Einstein relation for biased random walk on Galton-Watson trees
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2013
SP  - 698
EP  - 721
VL  - 49
IS  - 3
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/12-AIHP486/
DO  - 10.1214/12-AIHP486
LA  - en
ID  - AIHPB_2013__49_3_698_0
ER  - 
%0 Journal Article
%A Ben Arous, Gerard
%A Hu, Yueyun
%A Olla, Stefano
%A Zeitouni, Ofer
%T Einstein relation for biased random walk on Galton-Watson trees
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2013
%P 698-721
%V 49
%N 3
%I Gauthier-Villars
%U http://www.numdam.org/articles/10.1214/12-AIHP486/
%R 10.1214/12-AIHP486
%G en
%F AIHPB_2013__49_3_698_0
Ben Arous, Gerard; Hu, Yueyun; Olla, Stefano; Zeitouni, Ofer. Einstein relation for biased random walk on Galton-Watson trees. Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 3, pp. 698-721. doi : 10.1214/12-AIHP486. http://www.numdam.org/articles/10.1214/12-AIHP486/

[1] E. Aïdékon. Speed of the biased random walk on a Galton-Watson tree. Preprint, 2011. Available at arXiv:1111.4313v3.

[2] D. J. Aldous and A. Bandyopadhyay. A survey of max-type recursive distributional equations. Ann. Appl. Probab. 15 (2005) 1047-1110. | MR | Zbl

[3] A. Dembo, N. Gantert, Y. Peres and O. Zeitouni. Large deviations for random walks on Galton-Watson trees: Averaging and uncertainty. Probab. Theory Related Fields 122 (2002) 241-288. | MR | Zbl

[4] A. Dembo and J. D. Deuschel. Markovian perturbation, response and fluctuation dissipation theorem. Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010) 822-852. | Numdam | MR | Zbl

[5] A. Dembo and N. Sun. Central limit theorem for biased random walk on multi-type Galton-Watson trees. Electron. J. Probab. 17 (2012) 1-40. | MR | Zbl

[6] G. Faraud. A central limit theorem for random walk in random environment on marked Galton-Watson trees. Electron. J. Probab. 16 (2011) 174-215. | MR | Zbl

[7] G. Faraud, Y. Hu and Z. Shi. Almost sure convergence for stochastically biased random walks on trees. Probab. Theory Related Fields 154 (2012) 621-660. | MR | Zbl

[8] W. Feller. An Introduction to Probability Theory and Its Applications, 3rd edition. Wiley, New York, 1968. | MR | Zbl

[9] N. Gantert, P. Mathieu and A. Piatnitski. Einstein relation for reversible diffusions in random environments. Commun. Pure Appl. Math. 65 (2012) 187-228. | MR | Zbl

[10] T. Komorowsky and S. Olla. Einstein relation for random walks in random environments. Stochastic Process. Appl. 115 (2005) 1279-1301. | MR | Zbl

[11] T. Komorowsky and S. Olla. On mobility and Einstein relation for tracers in time-mixing random environments. J. Stat. Phys. 118 (2005) 407-435. | MR | Zbl

[12] J. L. Lebowitz and H. Rost. The Einstein relation for the displacement of a test particle in a random environment. Stochastic Process. Appl. 54 (1994) 183-196. | MR | Zbl

[13] M. Loulakis. Einstein relation for a tagged particle in simple exclusion processes. Comm. Math. Phys. 229 (2002) 347-367. | MR | Zbl

[14] R. Lyons. A simple path to Biggins' martingale convergence for branching random walk. In Classical and Modern Branching Processes (Minneapolis, MN, 1994) 217-221. IMA Vol. Math. Appl. 84. Springer, New York, 1997. | MR | Zbl

[15] R. Lyons. Random walks and percolation on trees. Ann. Probab. 18 (1990) 931-958. | MR | Zbl

[16] R. Lyons, R. Pemantle and Y. Peres. Ergodic theory on Galton-Watson trees: Speed of random walk and dimension of harmonic measure. Ergodic Theory Dynam. Systems 15 (1995) 593-619. | MR | Zbl

[17] R. Lyons, R. Pemantle and Y. Peres. Biased random walks on Galton-Watson trees. Probab. Theory Related Fields 106 (1996) 249-264. | MR | Zbl

[18] R. Pemantle and Y. Peres. The critical Ising model on trees, concave recursions and nonlinear capacity. Ann. Probab. 38 (2010) 184-206. | MR | Zbl

[19] Y. Peres and O. Zeitouni. A central limit theorem for biased random walks on Galton-Watson trees. Probab. Theory Related Fields 140 (2008) 595-629. | MR | Zbl

[20] O. Zeitouni. Random walks in random environment. In XXXI Summer School in Probability, St. Flour (2001) 193-312. Lecture Notes in Math. 1837. Springer, Berlin, 2004. | MR | Zbl

Cité par Sources :