Spectral gaps and exponential integrability of hitting times for linear diffusions

Loukianov, Oleg; Loukianova, Dasha; Song, Shiqi

Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 3, pp. 679-698.

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+=supx≥am(]x, +∞[)(S(x)-S(a)), Ba-=supx≤am(]-∞; 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.

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.

Classification : 60J25, 60J35, 60J60
Mots clés : recurrence, linear Markov process, exponential moments, hitting times, Poincaré inequality, spectral gap, Dirichlet form

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