Extremes of branching Ornstein-Uhlenbeck processes

In this article, we focus on the asymptotic behaviour of extremal particles in a branching Ornstein-Uhlenbeck process: particles move according to an Ornstein-Uhlenbeck process, solution of dXs = −µXsds + dBs, and branch at rate 1. We make µ = µt depend on the time-horizon t at which we observe the particles positions and we suppose that µtt → γ ∈ (0, ∞]. We show that, properly centred and normalised, the extremal point process continuously interpolates between the extremal point process of the branching Brownian motion (case γ = 0) and the extremal point process of independent Gaussian random variables (case γ = ∞). Along the way, we obtain several results on standard branching Brownian motion of intrinsic interest. In particular, we give a probabilistic representation of the main object of study in [DMS16] which is the probability that the maximal position has an abnormally high velocity