This talk concerns the rigorous derivation of diffusion, thermalization and decoherence in simple quantum systems: a spin or a particle interacting with phonons.
Our approach takes the Markovian description (Pauli master equation or quantum Boltzmann equation) as a starting point. However, since such a Markovian description is only valid in a
certain scaling limit (vanishing coupling constant and long time), we develop an expansion around the Markovian description to control the behavior at finite coupling constant.  This
expansion consists of a controlled renormalization group flow with Brownian motion as the fix-point solution.

The long-range nature of the velocity-velocity correlation function is an example of a physical quantity that is not described correctly within the Markovian description (cfr.
Alder-Wainwright), but that is captured by our approach.