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We consider Kallenberg's hypothesis on the characteristic function of aL\'{e}vy process and show that it allows the construction of weakly continuousbridges of the L\'{e}vy process conditioned to stay positive. We thereforeprovide a notion of normalized excursions L\'{e}vy processes above theircumulative minimum. Our main contribution is the construction of a continuousversion of the transition density of the L\'{e}vy process conditioned to staypositive 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 provideupper semi-continuous versions. Using the bridges of the conditioned L\'{e}vyprocess, the Durrett-Iglehart theorem stating that the Brownian bridge from $0$to $0$ conditioned to remain above $-\varepsilon$ converges weakly to theBrownian excursion as $\varepsilon \to0$, is extended to L\'{e}vy processes. Wealso extend the Denisov decomposition of Brownian motion to L\'{e}vy processesand their bridges, as well as Vervaat's classical result stating theequivalence in law of the Vervaat transform of a Brownian bridge and thenormalized Brownian excursion.

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