Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval

Consider a completely asymmetric Lévy process which has absolutely continuous transition probabilities. We determine the exponential decay parameter ρ and the quasistationary distribution for the transition probabilities of the Lévy process killed as it exits from a finite interval, prove that the killed process is ρ-positive and specify the ρ-invariant function and measure.