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Quasi-stationary distributions for Lévy processes. (English) Zbl 1130.60054
Let \(X=\{ X_{t}:t\geq 0\} \) be a Lévy process defined on a filtered space \(( \Omega ,\mathcal{F},\mathbf{F},P) \) where the filtration \(\mathbf{F}=\{ \mathcal{F}_{t}:t\geq 0\} \) is assumed to satisfy the usual assumptions of right continuity and completion. Consider the probabilities \(\{ P_{x}:x\in \mathbb{R}\} \) such that \(P_{x}( X_{0}=x) =1\) and \(P_{0}=P\). Define the first passage time into the lower half-line \(( -\infty ,0) \) by \(\tau =\inf \{t>0:X_{t}<0\} \). The main object of the authors’ interest is the existence and characterization of the so-called limiting quasi-stationary distribution (or Yaglom’s limit) \(\lim_{t\uparrow \infty}P_{x}( X_{t}\in B| \tau >t) =\mu ( B) \) for \(B\in \mathcal{B}( [0,\infty )) \) where, in particular, \(\mu \) does not depend on the initial state \(x\geq 0\). The authors consider this limit for \(x\geq 0\) in the case of Lévy processes for which \(( -\infty ,0) \) is irregular for \(0\), and for \(x>0\) in the case of Lévy processes for which \(( -\infty ,0) \) is regular for \(0\). The existence and characterization result obtained improves the analogous result for spectrally positive compound Poisson processes with negative drift and jump distribution with rational characteristic function proved (within the context of the \(M/G/1\) queue) by E. K. Kyprianou [J. Appl. Prob. 8, 494–507 (1971; Zbl 0234.60121)] as well as the result for Brownian motion with drift proved by S. Martinez and J. San Martin [J. Appl. Probab. 31, 911–920 (1994; Zbl 0818.60071)], and complements an analogous result for random walks obtained by D. L. Iglehart as well as related results due to R. A. Doney.

MSC:
60G51 Processes with independent increments; Lévy processes
60F05 Central limit and other weak theorems
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