Bridges of L\'{e}vy processes conditioned to stay positive

We consider Kallenberg's hypothesis on the characteristic function of a L\'{e}vy process and show that it allows the construction of weakly continuous bridges of the L\'{e}vy process conditioned to stay positive. We therefore provide a notion of normalized excursions L\'{e}vy processes above their cumulative minimum. Our main contribution is the construction of a continuous version of the transition density of the L\'{e}vy process conditioned to stay positive by using the weakly continuous bridges of the L\'{e}vy process itself. For this, we rely on a method due to Hunt which had only been shown to provide upper semi-continuous versions. Using the bridges of the conditioned L\'{e}vy process, the Durrett-Iglehart theorem stating that the Brownian bridge from $0$ to $0$ conditioned to remain above $-\varepsilon$ converges weakly to the Brownian excursion as $\varepsilon \to0$, is extended to L\'{e}vy processes. We also extend the Denisov decomposition of Brownian motion to L\'{e}vy processes and their bridges, as well as Vervaat's classical result stating the equivalence in law of the Vervaat transform of a Brownian bridge and the normalized Brownian excursion.