  Natural disasters may have considerable impact on society as well as on
(re)insurance industry. Max-stable processes are ideally suited for the
modeling of the spatial extent of such extreme events, but it is often assumed
that there is no temporal dependence. Only a few papers have introduced
spatio-temporal max-stable models, extending the Smith, Schlather and
Brown-Resnick spatial processes. These models suffer from two major drawbacks:
time plays a similar role as space and the temporal dynamics is not explicit.
In order to overcome these defects, we introduce spatio-temporal max-stable
models where we partly decouple the influence of time and space in their
spectral representations. We introduce both continuous and discrete-time
versions. We then consider particular Markovian cases with a max-autoregressive
representation and discuss their properties. Finally, we briefly propose an
inference methodology which is tested through a simulation study.
