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Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. (English) Zbl 0880.60077
The paper is concerned with a spectrally negative Lévy process killed as it exists from a finite interval. The author is interested in the properties of the transition function \(P^t(x,A)\) of this process as function of \(t\) and \(x\) (Sec. 3-5) and the decay and ergodic properties of this function (Sec. 6). In particular, it is proved that if the one-dimensional distribution of the original process is absolutely continuous (condition AC), then \(P^t\) is \(\rho\)-positive and \(\exp(\rho t)P^t(x,\cdot)\) converges as \(t\to\infty\) in the weak sense to some definite measure.

MSC:
60J99 Markov processes
28D10 One-parameter continuous families of measure-preserving transformations
60G50 Sums of independent random variables; random walks
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