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July 2015 Spectral gap for stochastic energy exchange model with nonuniformly positive rate function
Makiko Sasada
Ann. Probab. 43(4): 1663-1711 (July 2015). DOI: 10.1214/14-AOP916

Abstract

We give a lower bound on the spectral gap for a class of stochastic energy exchange models. In 2011, Grigo et al. introduced the model and showed that, for a class of stochastic energy exchange models with a uniformly positive rate function, the spectral gap of an $N$-component system is bounded from below by a function of order $N^{-2}$. In this paper, we consider the case where the rate function is not uniformly positive. For this case, the spectral gap depends not only on $N$ but also on the averaged energy $\mathcal{E}$, which is the conserved quantity under the dynamics. Under some assumption, we obtain a lower bound of the spectral gap which is of order $C(\mathcal{E})N^{-2}$ where $C(\mathcal{E})$ is a positive constant depending on $\mathcal{E}$. As a corollary of the result, a lower bound of the spectral gap for the mesoscopic energy exchange process of billiard lattice studied by Gaspard and Gilbert [ J. Stat. Mech. Theory Exp. 2008 (2008) p11021, J. Stat. Mech. Theory Exp. 2009 (2009) p08020] and the stick process studied by Feng et al. [ Stochastic Process. Appl. 66 (1997) 147–182] are obtained.

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Makiko Sasada. "Spectral gap for stochastic energy exchange model with nonuniformly positive rate function." Ann. Probab. 43 (4) 1663 - 1711, July 2015. https://doi.org/10.1214/14-AOP916

Information

Received: 1 July 2013; Revised: 1 January 2014; Published: July 2015
First available in Project Euclid: 3 June 2015

zbMATH: 1330.60120
MathSciNet: MR3353812
Digital Object Identifier: 10.1214/14-AOP916

Subjects:
Primary: 60K35
Secondary: 82C31

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.43 • No. 4 • July 2015
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