Lévy Processes Conditioned on Having a Large Height Process

In the present work, we consider spectrally positive Levy processes (X-t, t >= 0) not drifting to +oo 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 Levy processes to stay positive. The (honest) law P-x(star) of this conditioned process (starting at x > 0) is defined as a Doob h-transform via a martingale. For Levy processes with infinite variation paths, this martingale is (f /it (dz)eaz +)1{t t > 0) is the past infimum process of X, where (A,t >0) is the so-called exploration process defined in [10] and where To is the hitting time of 0 for X. Under P-x(star), we also obtain a path decomposition of X at its minimum, which enables us to prove the convergence of P-x(star) 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.