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20 Giugno, 2024 11:00
Sezione di Probabilità e Statistica Matematica

A new stochastic order applied to branching random walks

Daniela Bertacchi, Università di Milano-Bicocca

Stochastic ordering is a powerful tool to compare random variables and processes.
We introduce a new type of stochastic ordering which generalizes the germ order introduced by Hutchcroft in 2022, which uses the generating function of the process.
We apply this order to compare different branching random walks on a countable space $X$. We prove that given two branching random walks with law $\mu$ and $\nu$ respectively, with $\mu\ge\nu$, then in every set where there is extinction according to $\mu$, there is extinction also according to $\nu$. Moreover, in every set where there is strong local survival according to $\nu$, there is strong local survival also according to $\mu$, provided that the supremum of the global extinction probabilities, for the
$\nu$-process, taken over all starting points $x$, is strictly smaller than 1. New conditions for survival and strong survival for inhomogeneous branching random walks are provided.
We also extend a result of Moyal (1962), on the properties of fixed points of the generating function of a branching random walk.