Spectral gaps and exponential integrability of hitting times for linear diffusions

Oleg Loukianov; Dasha Loukianova; Shiqi Song

Annales de l'I.H.P. Probabilités et statistiques (2011)

  • Volume: 47, Issue: 3, page 679-698
  • ISSN: 0246-0203

Abstract

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Let X be a regular continuous positively recurrent Markov process with state space ℝ, scale function S and speed measure m. For a∈ℝ denote Ba+=supx≥am(]x, +∞[)(S(x)−S(a)), Ba−=supx≤am(]−∞; x[)(S(a)−S(x)). It is well known that the finiteness of Ba± is equivalent to the existence of spectral gaps of generators associated with X. We show how these quantities appear independently in the study of the exponential moments of hitting times of X. Then we establish a very direct relation between exponential moments and spectral gaps, all by improving their classical bounds.

How to cite

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Loukianov, Oleg, Loukianova, Dasha, and Song, Shiqi. "Spectral gaps and exponential integrability of hitting times for linear diffusions." Annales de l'I.H.P. Probabilités et statistiques 47.3 (2011): 679-698. <http://eudml.org/doc/243348>.

@article{Loukianov2011,
abstract = {Let X be a regular continuous positively recurrent Markov process with state space ℝ, scale function S and speed measure m. For a∈ℝ denote Ba+=supx≥am(]x, +∞[)(S(x)−S(a)), Ba−=supx≤am(]−∞; x[)(S(a)−S(x)). It is well known that the finiteness of Ba± is equivalent to the existence of spectral gaps of generators associated with X. We show how these quantities appear independently in the study of the exponential moments of hitting times of X. Then we establish a very direct relation between exponential moments and spectral gaps, all by improving their classical bounds.},
author = {Loukianov, Oleg, Loukianova, Dasha, Song, Shiqi},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {recurrence; linear Markov process; exponential moments; hitting times; Poincaré inequality; spectral gap; Dirichlet form},
language = {eng},
number = {3},
pages = {679-698},
publisher = {Gauthier-Villars},
title = {Spectral gaps and exponential integrability of hitting times for linear diffusions},
url = {http://eudml.org/doc/243348},
volume = {47},
year = {2011},
}

TY - JOUR
AU - Loukianov, Oleg
AU - Loukianova, Dasha
AU - Song, Shiqi
TI - Spectral gaps and exponential integrability of hitting times for linear diffusions
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2011
PB - Gauthier-Villars
VL - 47
IS - 3
SP - 679
EP - 698
AB - Let X be a regular continuous positively recurrent Markov process with state space ℝ, scale function S and speed measure m. For a∈ℝ denote Ba+=supx≥am(]x, +∞[)(S(x)−S(a)), Ba−=supx≤am(]−∞; x[)(S(a)−S(x)). It is well known that the finiteness of Ba± is equivalent to the existence of spectral gaps of generators associated with X. We show how these quantities appear independently in the study of the exponential moments of hitting times of X. Then we establish a very direct relation between exponential moments and spectral gaps, all by improving their classical bounds.
LA - eng
KW - recurrence; linear Markov process; exponential moments; hitting times; Poincaré inequality; spectral gap; Dirichlet form
UR - http://eudml.org/doc/243348
ER -

References

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