Abstract. 
I will first discuss heuristically the properties of the transfer of energy
in extended Hamiltonian systems described by local collisional dynamics.
Those systems model aerogels which are gels whose liquid components have
been removed and replaced by gas atoms. Next, I will discuss stochastic
models which approximate those deterministic systems. The stochastic
dynamics consist of tracer particles moving through an array of hot
scatterers describing the effect of heat baths at fixed temperatures.
Those models have the structure of Markov renewal
processes. I will show an explicit formula for the thermal conductivity
and an equation for the temperature profiles. Finally, I will discuss the
large deviations of the current. I will show that the large deviations
functional takes a form different from the one generally expected and
valid for instance in stochastic lattice gases. I will show that this
effect may be understood as a consequence of the fact that the energy is
transferred locally by the motion of free particles which, under
equilibrium conditions, may get arbitrarily low speed with sufficiently
large probability.