Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Spectral gaps and exponential integrability of hitting times for linear diffusions

Oleg Loukianov, Dasha Loukianova, and Shiqi Song

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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+=supxam(]x, +∞[)(S(x)−S(a)),

Ba=supxam(]−∞; 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.

Résumé

Soit X un processus de Markov récurrent positif à trajectoires continues et à valeurs dans ℝ. Soient S sa fonction d’échelle et m sa mesure de vitesse. Pour a∈ℝ notons

Ba+=supxam(]x, +∞[)(S(x)−S(a)),

Ba=supxam(]−∞; x[)(S(a)−S(x)).

Il est bien connu que la finitude de Ba± est équivalente à l’existence d’un trou spectral du générateur associé à X. Nous montrons comment ces quantités apparaissent d’une manière indépendante dans l’étude des temps d’atteinte de X. Ensuite nous établissons une relation directe entre les moments exponentiels et le trou spectral, en améliorant en plus leurs encadrements classiques.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 47, Number 3 (2011), 679-698.

Dates
First available in Project Euclid: 23 June 2011

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1308834855

Digital Object Identifier
doi:10.1214/10-AIHP380

Mathematical Reviews number (MathSciNet)
MR2841071

Zentralblatt MATH identifier
1233.60044

Subjects
Primary: 60J25: Continuous-time Markov processes on general state spaces 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60J60: Diffusion processes [See also 58J65]

Keywords
Recurrence Linear Markov process Exponential moments Hitting times Poincaré inequality Spectral gap Dirichlet form

Citation

Loukianov, Oleg; Loukianova, Dasha; Song, Shiqi. Spectral gaps and exponential integrability of hitting times for linear diffusions. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), no. 3, 679--698. doi:10.1214/10-AIHP380. https://projecteuclid.org/euclid.aihp/1308834855


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