We consider a process Z on the real line composed from a Lévy process and its exponentially tilted version killed with arbitrary rates and give an expression for the joint law of the supremum S, its time T, and the process Z(T+.)-S. This expression is in terms of the laws of the original and the tilted Lévy processes conditioned to stay negative and positive respectively.

The result is used to derive a new representation of stationary particle systems driven by Lévy processes. In particular, this implies that a max-stable process arising from Lévy processes admits a mixed moving maxima representation with spectral functions given by the conditioned Lévy processes.

This talk is based on:  http://arxiv.org/abs/1405.3443