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

TL;DR

This paper investigates the decay rates of large deviations for the maximum position in a reducible two-type branching Brownian motion, revealing phase transitions influenced by parameters like variance and branching rate.

Contribution

It provides a detailed analysis of the decay rate function for large deviations, highlighting phase transitions based on model parameters, extending previous convergence results.

Findings

Decay rate function exhibits phase transitions.

Decay probabilities decay exponentially with rate depending on parameters.

Results extend understanding of maximum position behavior in reducible branching Brownian motions.

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 in \cite{BeMa21} 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 $t\rightarrow\infty$, \[ {\mathbb P}(M_t\geq \theta m_t),\quad \theta>1. \] We shall show that the decay rate function exhibits phase transitions depending on certain relations between $\theta$, the variance of the underlying Brownian motion and the branching rate.