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Abstract:We study a critical multitype branching particle system in $\mathbb{R}^N$ with finite type space $\mathcal{K}={1,\dots,K}$. Particles of type $i$ move according to a symmetric $\alpha_i$-stable process and reproduce according to a critical offspring law with irreducible stochastic mean matrix. All lifetime distributions are non-arithmetic. The type-$1$ lifetime distribution has infinite mean and a regularly varying tail
$$ 1-F_1(t)\sim c_1t^{-\gamma}, \qquad 0<\gamma<1, $$
whereas the lifetime distributions of types $2,\dots,K$ satisfy the polynomial upper-tail bounds
$$ 1-F_i(t)\le C t^{-\eta_i}, \qquad i=2,\dots,K, \qquad \eta_i>1, \qquad \eta:=\min_{2\le i\le K}\eta_i. $$
The offspring mechanism satisfies a $(1+\beta)$-stable-domain regular-variation condition, with $\beta\in(0,1]$. The initial population is a Poisson random measure with intensity
$$ \Lambda=\sum_{i=1}^K a_i\,\lambda\otimes\delta_i, \qquad a_i\ge0, $$
and we set $A:=\sum_{i=1}^K a_i$. Under the space--lifetime condition
$$ \rho:=\left(\eta-1\right)\wedge\frac{N}{\alpha_1} > \frac{\gamma}{\beta}, $$
we prove that the system converges to a Poisson random measure with intensity $A\lambda\otimes\delta_1$. Thus, although the initial population may assign positive spatial intensity to every type, the limiting population is supported entirely on the infinite-mean type while preserving the aggregate initial spatial intensity $A\lambda$.

Submission history

From: Jose Hermenegildo Ramirez Gonzalez [view email]
[v1] Tue, 9 Jun 2026 23:14:59 UTC (46 KB)
[v2] Thu, 3 Sep 2026 21:38:43 UTC (40 KB)