  The recent contribution Dieker & Mikosch (2015) [1] obtained important
representations of max-stable stationary Brown-Resnick random fields $\zeta_Z$
with a spectral representation determined by a Gaussian process $Z$. With
motivations from \cite{DM} we derive for some general $Z$, representations for
$\zeta_Z$ via exponential tilting of $Z$. Our main findings concern a)
Dieker-Mikosch representations of max-stable processes, b) two-sided extensions
of stationary max-stable processes, c) inf-argmax representation of any
max-stable distribution, and d) new formulas for generalised Pickands
constants. Our applications include new conditions for the stationarity of
$\zeta_Z$, a characterisation of Gaussian random vectors and an alternative
proof of Kabluchko's characterisation of Gaussian processes with stationary
increments.
