I will discuss some recent (but modest) results showing the existence and slow mixing of a stationary chain of Hamiltonian oscillators subject to a heat bath. Such systems are used as simple models of heat conduction or energy transfer. Though the unlimite goal might be seen to under stand the "fourier" like law in this setting, I will be less ambitious. I will show that under some hypotheses, the chain posses a unique stationary state. Surprisingly, even these simple results require some delicate stochastic averaging. Furthermore, it is the existence of a stationary measure (not the uniquness) which is difficult. This is joint work with Martin Hairer.