Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Einstein relation for biased random walk on Galton–Watson trees

Gerard Ben Arous, Yueyun Hu, Stefano Olla, and Ofer Zeitouni

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Abstract

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.

Résumé

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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 49, Number 3 (2013), 698-721.

Dates
First available in Project Euclid: 2 July 2013

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1372772641

Digital Object Identifier
doi:10.1214/12-AIHP486

Mathematical Reviews number (MathSciNet)
MR3112431

Zentralblatt MATH identifier
1296.60266

Subjects
Primary: 60K37: Processes in random environments
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 82C44: Dynamics of disordered systems (random Ising systems, etc.)

Keywords
Galton–Watson tree Einstein relation Spine representation

Citation

Ben Arous, Gerard; Hu, Yueyun; Olla, Stefano; Zeitouni, Ofer. Einstein relation for biased random walk on Galton–Watson trees. Ann. Inst. H. Poincaré Probab. Statist. 49 (2013), no. 3, 698--721. doi:10.1214/12-AIHP486. https://projecteuclid.org/euclid.aihp/1372772641


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