## Electronic Communications in Probability

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

#### 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

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

Rights

#### 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

#### References

• Aurzada, Frank; Dereich, Steffen. Small deviations of general Lévy processes. Ann. Probab. 37 (2009), no. 5, 2066–2092.
• Bertoin, Jean. Lévy processes. Cambridge Tracts in Mathematics, 121. Cambridge University Press, Cambridge, 1996. x+265 pp. ISBN: 0-521-56243-0
• Bertoin, Jean. Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. Ann. Appl. Probab. 7 (1997), no. 1, 156–169.
• Chung, K. L. Doubly-Feller process with multiplicative functional. Seminar on stochastic processes, 1985 (Gainesville, Fla., 1985), 63–78, Progr. Probab. Statist., 12, Birkhäuser Boston, Boston, MA, 1986.
• Davies, E. Brian. Linear operators and their spectra. Cambridge Studies in Advanced Mathematics, 106. Cambridge University Press, Cambridge, 2007. xii+451 pp. ISBN: 978-0-521-86629-3; 0-521-86629-4
• Doney, Ronald A. Fluctuation theory for Lévy processes. Lectures from the 35th Summer School on Probability Theory held in Saint-Flour, July 6â€“23, 2005. Edited and with a foreword by Jean Picard. Lecture Notes in Mathematics, 1897. Springer, Berlin, 2007. x+147 pp. ISBN: 978-3-540-48510-0; 3-540-48510-4
• Hawkes, John. Potential theory of Lévy processes. Proc. London Math. Soc. (3) 38 (1979), no. 2, 335–352.
• Engel, Klaus-Jochen; Nagel, Rainer. One-parameter semigroups for linear evolution equations. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. Graduate Texts in Mathematics, 194. Springer-Verlag, New York, 2000. xxii+586 pp. ISBN: 0-387-98463-1
• Pistorius, M. R. On exit and ergodicity of the spectrally one-sided Lévy process reflected at its infimum. J. Theoret. Probab. 17 (2004), no. 1, 183–220.
• Schilling, René L.; Wang, Jian. Strong Feller continuity of Feller processes and semigroups. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 15 (2012), no. 2, 1250010, 28 pp.
• Suprun, V. N. The ruin problem and the resolvent of a killed independent increment process. (Russian) Ukrain. Mat. Å½. 28 (1976), no. 1, 53–61, 142.
• Tuominen, Pekka; Tweedie, Richard L. Exponential decay and ergodicity of general Markov processes and their discrete skeletons. Adv. in Appl. Probab. 11 (1979), no. 4, 784–803.