The Annals of Probability

Einstein relation and steady states for the random conductance model

Nina Gantert, Xiaoqin Guo, and Jan Nagel

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Abstract

We consider random walk among i.i.d., uniformly elliptic conductances on $\mathbb{Z}^{d}$, and prove the Einstein relation (see Theorem 1). It says that the derivative of the velocity of a biased walk as a function of the bias equals the diffusivity in equilibrium. For fixed bias, we show that there is an invariant measure for the environment seen from the particle. These invariant measures are often called steady states. The Einstein relation follows at least for $d\geq3$, from an expansion of the steady states as a function of the bias (see Theorem 2), which can be considered our main result. This expansion is proved for $d\geq3$. In contrast to Guo [Ann. Probab. 44 (2016) 324–359], we need not only convergence of the steady states, but an estimate on the rate of convergence (see Theorem 4).

Article information

Source
Ann. Probab., Volume 45, Number 4 (2017), 2533-2567.

Dates
Received: December 2015
Revised: April 2016
First available in Project Euclid: 11 August 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1502438434

Digital Object Identifier
doi:10.1214/16-AOP1119

Mathematical Reviews number (MathSciNet)
MR3693969

Zentralblatt MATH identifier
1385.60061

Subjects
Primary: 60K37: Processes in random environments 60K40: Other physical applications of random processes
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60G10: Stationary processes 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

Keywords
Random conductance model Einstein relation steady states

Citation

Gantert, Nina; Guo, Xiaoqin; Nagel, Jan. Einstein relation and steady states for the random conductance model. Ann. Probab. 45 (2017), no. 4, 2533--2567. doi:10.1214/16-AOP1119. https://projecteuclid.org/euclid.aop/1502438434


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