L\'evy processes conditioned on having a large height process

In the present work, we consider spectrally positive L'evy processes $(X_t,t\ge 0)$ not drifting to $+\mathrm{\infty }$ 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 L'evy processes to stay positive. The (honest) law $\text{pfl}$ of this conditioned process is defined as a Doob $h$-transform via a martingale. For L'evy processes with infinite variation paths, this martingale is $(\int\tilde\rt($\mathrm{d}$z)e^{\alpha z}+I_t)\2{t\leq T_0}$forsome$\alpha$andwhere$(I_t,t\geq0)$isthepastinfimumprocessof$X$,where$(\tilde\rt,t\geq0)$ is the so-called \emph{exploration process} defined in Duquesne, 2002, and where $T_0$ is the hitting time of 0 for $X$. Under $\text{pfl}$, we also obtain a path decomposition of $X$ at its minimum, which enables us to prove the convergence of $\text{pfl}$ as $x\to 0$. 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 \emph{splitting tree}. We also can give an alternative characterization of our conditioned process in the vein of spine decompositions.