Hawkes processes are a popular model for self-exciting phenomena, from earthquakes to finance. In this talk, I will first present them in a simple way, using a Poisson imbedding construction. I will then review what is known about their long-time behavior, through limit theorems for both linear and non-linear cases. The focus will be on three regimes that appear when the process has a long memory and the branching ratio gets close to or above one: the Nearly Unstable, the Weakly Critical, and the Supercritical Nearly Unstable Hawkes processes. These regimes have been studied qualitatively, but quantitative convergence results have been missing. I will explain how we obtain explicit convergence rates, relying on a coupling with a Brownian sheet, Fourier analysis, and a careful approximation of the absolute value function.
 

We study the optimal control of mean-field systems with heterogeneous and asymmetric interactions. This leads to considering a family of controlled Brownian diffusion processes with dynamics depending on the whole collection of marginal robability laws. We prove the well-posedness of such systems and define the control problem together with its related value function. We next prove a law invariance property for the value function which allows us to work on the set of collections of probability laws. We show that the value function satisfies a dynamic programming principle (DPP) on the flow of collections of probability measures. We also derive a chain rule for a class of regular functions along the flows of collections of marginal laws of diffusion processes. Combining the DPP and the chain rule, we prove that the value function is a viscosity solution of a Bellman dynamic programming equation in a L²-set of Wasserstein space-valued functions. This talk is based on a joint work with A. De Crescenzo, M. Fuhrman and H. Pham.