@article{ECP3006,
author = {Martin Kolb and Mladen Savov},
title = {Exponential ergodicity of killed Lévy processes in a finite interval},
journal = {Electron. Commun. Probab.},
fjournal = {Electronic Communications in Probability},
volume = {19},
year = {2014},
keywords = {Markov processes; Levy processes; ergodicity; Banach spaces;},
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.
},
pages = {no. 31, 1-9},
issn = {1083-589X},
doi = {10.1214/ECP.v19-3006},
url = {http://ecp.ejpecp.org/article/view/3006}}