Maxima of stationary systems of randomly time-changed L\'evy particles

We construct stationary max-infinitely divisible (max-id) processes from systems of randomly time-changed L\'evy particles. Classical examples without time change, such as the Brown-Resnick process, are, up to marginal transformations, max-stable. We show that random time change of the underlying particles alters the dependence structure of the max-id process and leads, in general, beyond the max-stable setting. At the same time, stationarity is preserved by a suitable reconfiguration of the starting points of the particle system. We then prove that the extremal behavior of the resulting max-id process is linked to an associated max-stable L\'evy-Brown-Resnick process through the max-domain of attraction (MDA). Thus, our work combines potential theory for Markov processes and extreme value theory to yield a large class of new, non-trivial stationary processes in the MDA of a given L\'evy-Brown-Resnick process. So far, specific examples of processes in the MDA are scarce in the literature.