Abstract
Following Bertoin who considered the ergodicity and exponential decay of Levy processes in a finite domain, we consider general Levy 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 Levy 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 Levy 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.
Highlights
Introduction and resultsIn this short note we investigate the ergodic properties of general Lévy processes killed upon exiting a finite interval
Ta = inf{t > 0 : Xt ∈/ (0, a)}, a > 0 and X a Lévy process we study from spectral-theoretical point of view the killed process P (Xt ∈ ., Ta > t)
E.g. absolute continuity of the transition semigroup of the unkilled Lévy process, we prove that the killed semigroup is a compact operator
Summary
In this short note we investigate the ergodic properties of general Lévy processes killed upon exiting a finite interval Exit from such domains is known as the "doublesided exit problem". I.e. the doubly Feller property of the underlying Lévy process, we derive and discuss the exponential ergodicity of the semigroup of the killed Lévy process in the general case, i.e. when our Lévy process can make both positive and negative jumps. We connect this topic to the general theory of semigroups and explicitly demonstrate how the main result can be related to general spectral theory. The note is organized as follows: in the first section we introduce the notation and the main results; in the second section we discuss the implications of our results, their conditions and how far they can be extended, we point out some challenges; in the third section we provide the proof of our results
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