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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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