Electronic Communications in Probability

Exponential ergodicity of killed Lévy processes in a finite interval

Martin Kolb and Mladen Savov

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Markov processes Levy processes ergodicity Banach spaces

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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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