Quasi-stationary distributions for Lévy processes

In recent years there has been some focus on the behaviour of one dimensional Lévy processes and random walks conditioned to stay positive; see for example Bertoin (1993, 1996), Bertoin and Doney (1994), Chaumont (1996) and Chaumont and Doney (2004). The resulting conditioned process is transient. In older literature however, one encounters for special classes of random walks and Lévy processes a similar, but nonetheless different, type of asymptotic conditioning to stay positive which results in a limiting quasi-stationary distribution. We continue this theme into the general setting of a Lévy process fulfilling certain types of conditions which are analogues of known classes in the random walk literature. Our results generalize those of Kyprianou (1971) for special types of one sided compound Poisson processes with drift and Martinez and San Martin (1994) for Brownian motion with drift and complement the results of Iglehart (1974) and Doney (1989) and Bertoin and Doney (1996) for random walks.