  Non-equilibrium steady states are subject to intense investigations but still
poorly understood. For instance, the derivation of Fourier law in Hamiltonian
systems is a problem that still poses several obstacles. In order to
investigate non-equilibrium systems, stochastic models of energy-exchange have
been introduced and they have been used to identify universal properties of
non-equilibrium. In these notes, after a brief review of the problem of
anomalous transport in 1-dimensional Hamiltonian systems, some boundary-driven
interacting random systems are considered and the "duality approach" to their
rigorous mathematical treatment is reviewed. Duality theory, of which a brief
introduction is given, is a powerful technique to deal with Markov processes
and interacting particle systems. The content of these notes is mainly based on
the papers [10, 11, 12].
