Einstein relation for reversible diffusions in a random environment

Nina Gantert (1) , Pierre Mathieu (2) , Andrey Piatnitski (3, 4)

1 Lehrstuhl fur Warscheinlichkeitstheorie
2 LATP - Laboratoire d'Analyse, Topologie, Probabilités
3 Narvik University College
4 LPI RAS - P. N. Lebedev Physical Institute of the Russian Academy of Sciences [Moscow]

We consider reversible diffusions in random environment and prove the Einstein relation for this model. It says that the derivative of the effective velocity under an additional local drift equals the diffusivity of the model without drift. The Einstein relation is conjectured to hold for a variety of models but is proved insofar only in particular cases. Our proof makes use of homogenization arguments, the Girsanov transform, and a refinement of the regeneration times introduced by Lian Shen in [20].

Mots clés: MARKOV-PROCESSES; RANDOM-WALK; INVARIANCE-PRINCIPLE; MOTT LAW; HOMOGENIZATION; PARTICLE

Domaines: Probabilités [math.PR]

Nina Gantert, Pierre Mathieu, Andrey Piatnitski. Einstein relation for reversible diffusions in a random environment. Communications on Pure and Applied Mathematics, 2012, 65 (2), pp.187-228. ⟨10.1002/cpa.20389⟩. ⟨hal-01271048⟩