Large deviations for the maximum of a reducible two-type branching Brownian motion

We consider a two-type reducible branching Brownian motion, defined as a particle system on the real line in which particles of two types move according to independent Brownian motions and create offspring at a constant rate. Particles of type 1 can give birth to particles of types 1 and 2, but particles of type 2 only give birth to descendants of type 2. Under some specific conditions, Belloum and Mallein in BeMa21 show that the maximum position Mₜ of all particles alive at time t, suitably centred by a deterministic function mₜ, converge weakly. In this work, we are interested in the decay rate of the following upper large deviation probability, as t, \ P (Mₜ mₜ), >1. \ We shall show that the decay rate function exhibits phase transitions depending on certain relations between, the variance of the underling Brownian motion and the branching rate.