### Exponential ergodicity of killed Lévy processes in a finite interval

**Martin Kolb**

*(University of Paderborn)*

**Mladen Svetoslavov Savov**

*(The University of Reading)*

#### Abstract

Following Bertoin who considered the ergodicity and exponential decay of Lévy processes in a finite domain, we consider general Lévy processes and their ergodicity and exponential decay in a finite interval. More precisely, given $T_a=\inf\{t>0:\,X_t\notin (0,a)\}$, $a>0$ and $X$ a Levy process then we study from spectral-theoretical point of view the killed semigroup $P \left(X_t \in . ; T_a > t\right)$. Under general conditions, e.g. absolute continuity of the transition semigroup of the unkilled Lévy process, we prove that the killed semigroup is a compact operator. Thus, we prove stronger results in view of the exponential ergodicity and estimates of the speed of convergence. Our results are presented in a Lévy processes setting but are well applicable for Markov processes in a finite interval under information about Lebesgue irreducibility of the killed semigroup and that the killed process is a double Feller process. For example, this scheme is applicable to a work of Pistorius.

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Pages: 1-9

Publication Date: May 24, 2014

DOI: 10.1214/ECP.v19-3006

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