  Let $X(t),t\in \mathbb{R}$ be a stochastically continuous stationary
max-stable process with Fr\'{e}chet marginals $\Phi_\alpha, \alpha>0$ and set
$M_X(T)=\sup_{t \in [0,T]} X(t),T>0$. In the light of the seminal articles
[1,2], it follows that $A_T=M_X(T)/T^{1/\alpha}$ converges in distribution as
$T\to \infty$ to $\mathcal{H}_Z^{1/\alpha} X(1)$, where $\mathcal{H}_Z$ is the
Pickands constant corresponding to the spectral process $Z$ of $X$. In this
contribution we derive explicit formulas for $\mathcal{H}_Z$ in terms of $Z$
and show necessary and sufficient conditions for its positivity. From our
analysis it follows that $A_T^\beta,T>0$ is uniformly integrable for any $\beta
\in (0,\alpha)$. Further, we discuss the dissipative Rosi\'nski (or mixed
moving maxima) representation of $X$. Additionally, for Brown-Resnick $X$ we
show the validity of the celebrated Slepian inequality and obtain lower bounds
on the growth of supremum of Gaussian processes with stationary increments by
exploiting the link between Pickands constants and Wills functional. Moreover,
we derive upper bounds for supremum of centered Gaussian processes given in
terms of Wills functional, and discuss the relation between Pickands and
Piterbarg constants.
