Electronic Communications in Probability

Exponential ergodicity of killed Lévy processes in a finite interval

Martin Kolb and Mladen Savov

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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 $. 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.<br />

Article information

Source
Electron. Commun. Probab., Volume 19 (2014), paper no. 31, 9 pp.

Dates
Accepted: 24 May 2014
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465316733

Digital Object Identifier
doi:10.1214/ECP.v19-3006

Mathematical Reviews number (MathSciNet)
MR3216565

Zentralblatt MATH identifier
1312.60057

Subjects
Primary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60G51 47D99 47A10: Spectrum, resolvent 37A25: Ergodicity, mixing, rates of mixing

Keywords
Markov processes Levy processes ergodicity Banach spaces

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Kolb, Martin; Savov, Mladen. Exponential ergodicity of killed Lévy processes in a finite interval. Electron. Commun. Probab. 19 (2014), paper no. 31, 9 pp. doi:10.1214/ECP.v19-3006. https://projecteuclid.org/euclid.ecp/1465316733


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