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

Makiko Sasada

Research output: Contribution to journalArticle

7 Citations (Scopus)

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 languageEnglish
Pages (from-to)1663-1711
Number of pages49
JournalAnnals of Probability
Volume43
Issue number4
DOIs
Publication statusPublished - 2015

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Rate Function
Spectral Gap
Energy
Lower bound
Model
Conserved Quantity
Billiards
Stochastic Processes
Corollary
Lower bounds
Class

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

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

In: Annals of Probability, Vol. 43, No. 4, 2015, p. 1663-1711.

Research output: Contribution to journalArticle

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