Mixing in deterministic and random PDE: lecture notes for the Santa Barbara Kinetic summer school

The perturbation Γ may be either random or deterministic. We will focus on two complementary situations. The first regime is when L is very mixing but non-dissipative, while Γ is dissipative. Their interaction leads to a dramatically increased dissipation, that can not be achieved by either of them alone. This phenomenon is known as relaxation enhancement. The second set of problems involves random perturbations Γ of a simple deterministic background operator L. Here, the long time dynamics leads to diffusive limits that are a result of the interaction of the deterministic dynamics generated by L and the random dynamics coming from Γ. We will look at several examples of such diffusive behavior, in the order of increasing difficulty. In this introductory section, we will consider two simple examples of each of these two classes of problems, just to illustrate what one may expect.