Description
Propagation and generation of chaos is an important ingredient for rigorous control of applicability of kinetic theory, in general. Chaos is here understood as sufficient statistical independence of random variables related to the "kinetic" observables of the system. In these lectures, we discuss how cumulants can be used to control the magnitude of such independence, using our recent work with Aleksis Vuoksenmaa on the stochastic Kac model [arxiv.org:2407.17068] as a benchmark. We introduce cumulants and Wick polynomials of random variables, and show how they can be used to generate evolution hierarchies for both deterministic and stochastic dynamical systems. Cumulants can measure weak joint dependence accurately, and we explain how this allows controlling the accuracy of kinetic theory in the stochastic Kac model. In that case, almost arbitrary initial data may be studied and we find that the evolution of the system is divided into three regimes: an initial regime of duration of at most O(ln N) for N particles in which the state of the system becomes chaotic, the kinetic regime which is determined by solutions to the Boltzmann-Kac equation, and an equilibrium regime. The techniques involve finding suitable weighted norms and iterative structures for the cumulant hierarchy, such as introducing partition classifiers to index the cumulants.