Extinction probabilities in branching processes with countably many types: a general framework

We consider Galton-Watson branching processes with countable typeset $\mathcal{X}$. study the vectors ${\bf q}(A)=(q_x(A))_{x\in\mathcal{X}}$ recording conditional probabilities of extinction in subsets types $A\subseteq \mathcal{X}$, given that type initial individual is $x$. first investigate location q}(A)$ set fixed points progeny generating vector and prove $q_x(\{x\})$ larger than or equal to $x$th entry any point, whenever it different from 1. Next, we present equivalent conditions for $q_x(A)< q_x (B)$ $x$ $A,B\subseteq \mathcal{X}$. Finally, develop a general framework characterise all \emph{distinct} probability vectors, thereby determine whether there are finitely many, countably uncountably many distinct vectors. illustrate our results examples, conclude open questions.