the continuity of $Y$ that there exist constants $C,\alpha,\delta > 0$, such that $$\begin{aligned}
    \sE |Y(s) - Y(t)|^2 \leq \frac{C}{|\log \|s-t\| |^{1+\alpha}}
  \end{aligned}$$ for all $s,t\in{\mathbb R}^d$ with $\|s-t\|<\delta$. Further let $\gamma(t) =
  \sE(Y(t) - Y(0))^2$ and $\sigma^2(t) = \sE(Y(t))^2$, $t \in {\mathbb R}^d$, denote the variogram and the variance of $Y$, respectively. Then, with a Fréchet point process $\sum_{i\in{\mathbb N}} \delta_{U_i}$ and independent copies $Y_i$ of $Y$, $i\in{\mathbb N}$, the process $$\begin{aligned}
    \label{BR_proc}
    \xi(t) = \max_{i\in{\mathbb N}} U_i \exp\left(Y_i(t) - \sigma^2(t) / 2\right), 
    \quad t\in{\mathbb R}^d,
  \end{aligned}$$ is stationary and its distribution only depends on the variogram $\gamma$. Comparing with the incremental representation , the distribution of the increments is given by the log-Gaussian random field $W(t) = \exp\left(Y(t) -
  \sigma^2(t) / 2\right)$, $t\in{\mathbb R}^d$, and Theorem \[theo\_cond\_increments\_cont\] applies.

Mixed moving maxima representation {#M3representation}
==================================

A large and commonly used class of max-stable processes is the class of M3 processes . Let $$\begin{aligned}
   \label{pi0}
   \Pi_0 = \sum_{i\in{\mathbb N}} \delta_{(U_i.T_i,F_i)}
 \end{aligned}$$ be the corresponding PPP on $(0,\infty)\times{\mathbb R}^d\times C({\mathbb R}^d)$ with intensity $u^{-2}\rd u \,\rd t \,\sP_F(\rd f)$. In the sequel, M3 processes are denoted by $$\begin{aligned}
  M(t) = \max_{i\in{\mathbb N}} U_i F_i(t- T_i), \quad t\in{\mathbb R}^d.\end{aligned}$$ The marginal distributions 