Lévy Process Conditioned by Its Height Process

In the present work, we consider spectrally positive Lévy processes (Xt, t ≥ 0) not drifting to +∞ and we are interested in conditioning these processes to reach arbitrarily large heights (in the sense of the height process associated with X) before hitting 0. This way we obtain a new conditioning of Lévy processes to stay positive. The (honest) law Px of this conditioned process is defined as a Doob h-transform via a martingale. For Lévy processes with infinite variation paths, this martingale is (∫ ρ̃t(dz)e αz + It ) 1{t≤T0} for some α and where (It, t ≥ 0) is the past infimum process of X, where (ρ̃t, t ≥ 0) is the socalled exploration process defined in [10] and where T0 is the hitting time of 0 for X. Under Px, we also obtain a path decomposition of X at its minimum, which enables us to prove the convergence of Px as x→ 0. When the process X is a compensated compound Poisson process, the previous martingale is defined through the jumps of the future infimum process of X. The computations are easier in this case because X can be viewed as the contour process of a (sub)critical splitting tree. We also can give an alternative characterization of our conditioned process in the vein of spine decompositions.