Einstein relation for reversible diffusions in random environment

TL;DR

This paper proves the Einstein relation for reversible diffusions in random environments, showing that the derivative of effective velocity equals diffusivity, using homogenization, Girsanov transform, and regeneration times.

Contribution

It provides a rigorous proof of the Einstein relation for a class of reversible diffusions in random environments, extending prior partial results.

Findings

Einstein relation holds for reversible diffusions in random environments.

Effective velocity derivative equals diffusivity in the studied model.

Homogenization and regeneration techniques are key to the proof.

Abstract

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].