Apparaît dans la collection : 2017 - T2 - Stochastic dynamics out of equilibrium

I will discuss how noise can arise in deterministic systems with strong instability with respect to the initial conditions. Starting with a discussion of the Central limit theorem for deterministic systems with random initial conditions and trying to get to the derivation of stochastic processes as limit theorems. Lesson 1 - Chaos and randomnessI will describe how establish a CLT or an invariance principle for observables of a deterministic system. To avoid technical details I will describe various approaches in the simplest possible case: smooth expanding maps. Lesson 2 - Gaussian or not GaussianI will discuss the large deviations for a deterministic system and discuss (again in the simplest possible setting) how to compute the rate function and its relations with the rate function of a Gaussian process. Also I will briefly comment on the class of systems to which the previous results can be generalised. Lesson 3 - Non-equilibriumI will discuss (deterministic) systems with a locally (almost) conserved quantity, explain how averaging theory applies and discuss how to prove that the evolution of the almost conserved quantity might be very close to the solution of an SDE for a very long time.