An equivalent representation of the Brown–Resnick process

Brown and Resnick (1977) introduce a max-stable process that is obtained as a limit of maxima of independent Ornstein–Uhlenbeck processes. As shown in Kabluchko et al. (2009) this process is dissipative and it therefore admits a mixed moving maxima representation. We show that the distribution of the spectral functions in this representation equals a well-known diffusion, namely a standard Brownian motion with drift conditional on taking negative values only. This can be used for fast simulation methods.