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