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.