The recent contribution Dieker & Mikosch (2015) [1] obtained important representations of max-stable stationary Brown-Resnick random fields $ζ_Z$ with a spectral representation determined by a Gaussian process $Z$. With motivations from \cite{DM} we derive for some general $Z$, representations for $ζ_Z$ 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 $ζ_Z$, a characterisation of Gaussian random vectors and an alternative proof of Kabluchko's characterisation of Gaussian processes with stationary increments.