= Spectral survival criterion for multitype Poisson branching
{title2=$\sum_i p_i>0\iff\rho(\Lambda)>1$}
This criterion holds even for reducible <nonnegative matrices>. If a <fixed point> $p\ne0$ exists, $1-e^{-z}<z$ for $z>0$ gives $(\Lambda p)_i>p_i$ on its support. Some $\alpha>1$ therefore satisfies $\Lambda p\ge\alpha p$, forcing the <spectral radius> above one by the <Perron–Frobenius theorem>. Conversely take $v\ge0$, $v\ne0$, with $\Lambda v=\rho v$ and $\rho>1$. For sufficiently small $\eta>0$, $1-e^{-\eta\rho v_i}\ge\eta v_i$. Thus $\eta v$ is a subsolution; monotone iteration under $T$ gives a positive <fixed point> below $\mathbf1$. Strictness of $1-e^{-z}<z$ excludes survival at the critical value for this Poisson offspring law.
Back to article page
