Parisian quasi-stationary distributions for asymmetric Lévy processes

In recent years there has been some focus on quasi-stationary behavior of an one-dimensional Lévy process X, where we ask for the law P(Xt∈dy|τ0−>t) for t→∞ and τ0−=inf{t≥0:Xt<0}. In this paper we address the same question for so-called Parisian ruin time τθ, that happens when process stays below zero longer than independent exponential random variable with intensity θ.