A Levy-derived process seen from its supremum and max-stable processes

We consider a process Z on the real line composed from a Levy process and its exponentially tilted version killed with arbitrary rates and give an expression for the joint law of the supremum (Z) over bar, its time T, and the process Z(T + center dot) - (Z) over bar. This expression is in terms of the laws of the original and the tilted Levy processes conditioned to stay negative and positive respectively. The result is used to derive a new representation of stationary particle systems driven by Levy processes. In particular, this implies that a max-stable process arising from Levy processes admits a mixed moving maxima representation with spectral functions given by the conditioned Levy processes.