Large Deviations for the Maximum of a Reducible Two-type Branching Brownian Motion

Abstract

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 showed that the maximum position M_t of all particles alive at time t, suitably centered by a deterministic function m_t, converge weakly. In this work, we are interested in the decay rate of the following upper large deviation probability, as t → ∞

$$\mathbb{P}(M_{t}\geq \theta m_{t}), \quad \theta >1.$$

We shall show that the decay rate function exhibits phase transitions depending on certain relations between θ, the variance of the underlying Brownian motion and the branching rate.