On the Structure and Representations of Max--Stable Processes

We develop classification results for max--stable processes, based on their spectral representations. The structure of max--linear isometries and minimal spectral representations play important roles. We propose a general classification strategy for measurable max--stable processes based on the notion of co--spectral functions. In particular, we discuss the spectrally continuous--discrete, the conservative--dissipative, and positive--null decompositions. For stationary max--stable processes, the latter two decompositions arise from connections to non--singular flows and are closely related to the classification of stationary sum--stable processes. The interplay between the introduced decompositions of max--stable processes is further explored. As an example, the Brown-Resnick stationary processes, driven by fractional Brownian motions, are shown to be dissipative. A result on general Gaussian processes with stationary increments and continuous paths is obtained.