Double jump in the maximum of two-type reducible branching Brownian motion

Abstract

Consider a two-type reducible branching Brownian motion in which particles' diffusion coefficients and branching rates are influenced by their types. Here reducible means that type 1 particles can produce particles of type 1 and type 2, but type 2 particles can only produce particles of type 2. The maximum of this process is determined by two parameters: the ratio of the diffusion coefficients and the ratio of the branching rates for particles of different types. Belloum and Mallein [Electron. J. Probab. 26(2021), no. 61] identified three phases of the maximum and the extremal process, corresponding to three regions in the parameter space. We investigate how the extremal process behaves asymptotically when the parameters lie on the boundaries between these regions. An interesting phenomenon is that a double jump occurs in the maximum when the parameters cross the boundary of the so called anomalous spreading region, while only single jump occurs when the parameters cross the boundary between the remaining two regions.