### 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<sup>-2</sup>. 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 e{open}, 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(e{open})N<sup>-2</sup> where C(e{open}) is a positive constant depending on e{open}. 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.

Original language | English |
---|---|

Pages (from-to) | 1663-1711 |

Number of pages | 49 |

Journal | Annals of Probability |

Volume | 43 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2015 |

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### Keywords

- Energy exchange
- Locally confined hard balls
- Nonuniformly positive rate function
- Spectral gap

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Annals of Probability*,

*43*(4), 1663-1711. https://doi.org/10.1214/14-AOP916

**Spectral gap for stochastic energy exchange model with nonuniformly positive rate function.** / Sasada, Makiko.

Research output: Contribution to journal › Article

*Annals of Probability*, vol. 43, no. 4, pp. 1663-1711. https://doi.org/10.1214/14-AOP916

}

TY - JOUR

T1 - Spectral gap for stochastic energy exchange model with nonuniformly positive rate function

AU - Sasada, Makiko

PY - 2015

Y1 - 2015

N2 - 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 e{open}, 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(e{open})N-2 where C(e{open}) is a positive constant depending on e{open}. 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.

AB - 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 e{open}, 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(e{open})N-2 where C(e{open}) is a positive constant depending on e{open}. 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.

KW - Energy exchange

KW - Locally confined hard balls

KW - Nonuniformly positive rate function

KW - Spectral gap

UR - http://www.scopus.com/inward/record.url?scp=84930837061&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84930837061&partnerID=8YFLogxK

U2 - 10.1214/14-AOP916

DO - 10.1214/14-AOP916

M3 - Article

VL - 43

SP - 1663

EP - 1711

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 4

ER -