Path decompositions for real Levy processes
  Let $X$ be a real L\'evy process and let $\Xpos $ be the process conditioned to stay positive. We assume that $ 0 $ is regular for $(-\infty, 0)$ and $(0, +\infty) $ with respect to $X$. Using elementary excursion theory arguments, we provide a simple probabilistic description of the reversed paths of $X$ and $\Xpos $ at their first hitting time of $ (x, +\infty)$ and last passage time of $ (-\infty, x ] $, on a fixed time interval $[0, t]$, for a positive level $x$. From these reversion formulas, we derive an extension to general L\'evy processes of Williams' decomposition theorems, Bismut's decomposition of the excursion above the infimum and also several relations involving the reversed excursion under the maximum. 
Thomas Duquesne