A deduction of Fourier's law of heat transport from microscopic deterministic Hamiltonian dynamics is still missing, even on a heuristic level. Furthermore, divergence of the thermal conductivity is observed numerically in some one-dimensional system of oscillators. In physics literature often these problems are connected to the chaoticity (or the lack of it).

I will present some recent rigorous mathematical results on a chain of oscillators whose Hamiltonian dynamics is perturbed by a (local) noise term that conserves total energy (and possibly total momentum). In finite systems this noise destroys all integrals of the motion except energy (and possibly momentum), so these systems are nicely ergodic on shells of constant energy (and possibly momentum).

The decay of the energy current time correlation function C(t) can be estimated. In the case where only energy is conserved, thermal conductivity (TC) always remains finite in the limit of infinite system size, but when also momentum is conserved TC may diverge in dimensions 1 and 2. In the harmonic case C(t) can be computed explicitly and we prove that thermal conductivity is finite for d>2 or if an on-site potential is present, while it diverges if d=1 or 2.

In the case of harmonic interactions, we obtain the convergence of the Wigner distribution of the energy to a phonon Boltzman equation in the proper limit, and the thermal conductivity can then be computed from this kinetic equation.

In the case that only energy is conserved and all interactions are harmonic, Fourier's law can be obtained from the stationary non-equilibrium states, through an entropy production argument.

In the case of an-harmonic interactions, the microscopic deduction of Fourier's law is still a very difficult mathematical problem, even in the presence of noise. I will present a possible mathematical approach, based on the so-called ''non-gradient'' method used in hydrodynamic limits, and show some surprising numerical results in the one- dimensional case.