Quantitative Einstein relation for reversible diffusions in a random environment
Probability
2026-05-26 v1 Analysis of PDEs
Abstract
The Einstein relation describes the response of a diffusing particle to a small constant external force. It states that, as the force tends to zero, the ratio of the limiting velocity to the force magnitude converges to the diffusivity matrix of the unforced particle, evaluated in the force direction. Gantert, Mathieu, and Piatnitski (2012) proved this identity for reversible diffusions in random environments. We prove a quantitative version, with an explicit quenched algebraic rate.
Cite
@article{arxiv.2605.26082,
title = {Quantitative Einstein relation for reversible diffusions in a random environment},
author = {Ahmed Bou-Rabee and Ruizhe Xu},
journal= {arXiv preprint arXiv:2605.26082},
year = {2026}
}
Comments
52 pages, 1 figure