Exponential Ergodicity of killed Levy processes in a finite interval

Following Bertoin who considered the ergodicity and exponential decay of Levy processes in a finite domain [4], 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 is not an element of(0, a) }, a > 0 and X a Levy process then we study from spectral-theoretical point of view the killed process P (X-t is an element of., T-a > t). 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 once one can establish Lebesgue irreducibility of the killed semigroup and that the killed process is a doubly Feller process. For example, this scheme is applicable to the work of Pistorius [10].