We study the connections existing between max-infinitely divisible distributions and Poisson processes from the point of view of functional analysis. More precisely, we derive functional identities for the former by using well-known results of Poisson stochastic analysis. We also introduce a family of Markov semigroups whose stationary measures are the so-called multivariate max-stable distributions. Their generators thus provide a functional characterization of extreme valued distributions in any dimension. Additionally, we give a few functional identities associated to those semi-groups, namely a Poincar{é} identity and commutation relations. Finally, we present a stochastic process whose semigroup corresponds to the one we introduced and that can be expressed using extremal stochastic integrals.