Extremal process for irreducible multi-type branching Brownian motion (Q6634811)

In the paper authors define the distribution function \N\[\Nh(u, t):=\mathbb{P}(M_t\leqslant x),\,t\geqslant0,\,x\in\mathbb{R}, \N\]\Nwhich solves the F-KPP equation \N\[\Nu_t=\frac{1}{2}u_{xx}+u^2-u\N\]\Nwith the Heaviside initial condition \(u(0, x) = \mathbf{1}_{[0,\infty)}(x)\). The random process \(M_t\) is defined as ``the right-most position among all the particles alive at time \(t\)'', where the quantity of particles in time is described as follows: ``Initially, there is a particle at the origin and the particle moves according to a standard Brownian motion. After an exponential time with parameter \(1\), this particle dies and splits into \(2\) particles. The offspring move independently according to standard Brownian motion from the place they are born and obey the same branching mechanism as their parent.''\N\NThen, authors define the (irreducible) multitype branching Brownian motions. This is done as follows: ``Initially, there is a particle of type \(i\) at site \(x\) and it moves according a standard Brownian motion. After an exponential time with parameter \(a_i\), it dies and splits into \(k_1\) offspring of type \(1\), \(k_2\) offspring of type \(2,\dots, k_d\) offspring of type \(d\) with probability \(p_k(i)\), where \(k=(k_1,\dots, k_d)^T\). The offspring evolve independently, each moves according to a standard Brownian motion and each type \(j\) particle reproduces with law \(\{p_k(j):k\in\mathbb{N}^d\}\) after an exponential distributed lifetime with parameter \(a_j\). This procedure goes on.''\N\NThen, the article notices that the described multi-type branching Brownian motion is related to the certain system of F-KPP equations, whose solution is denoted by \(\mathbf{u}\) and \(\mathbf{v}:=\mathbf{1}-\mathbf{u}\).\N\NThe main results of the paper are described as follows ``Our first main result is on the convergence of \(\mathbf{v}\) to the traveling wave solution for a class of initial value conditions. Our second main result is about the characterization of the extremal process of multi-type branching Brownian motion.''