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Critical parameters of germ-monotone families of branching random walks

Daniela Bertacchi, Fabio Zucca

Abstract

We introduce a broad class of families of branching random walks on a set $X$. The processes in each family are parametrized by a positive parameter $λ$ and they are monotonically increasing in $λ$ with respect to the germ order, a notion that extends classical stochastic domination. We define a general notion of critical parameter $λ(A)$ associated with a subset $A \subseteq X$ and investigate how modifications of reproduction laws affect these critical parameters.

Critical parameters of germ-monotone families of branching random walks

Abstract

We introduce a broad class of families of branching random walks on a set . The processes in each family are parametrized by a positive parameter and they are monotonically increasing in with respect to the germ order, a notion that extends classical stochastic domination. We define a general notion of critical parameter associated with a subset and investigate how modifications of reproduction laws affect these critical parameters.
Paper Structure (10 sections, 18 theorems, 19 equations, 2 tables)

This paper contains 10 sections, 18 theorems, 19 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Basic definitions and preliminaries
  3. Discrete-time Branching Random Walks and their survival/extinction.
  4. Equivalent sets and comparison of extinction probability vectors
  5. Germ-monotone families of Branching Random Walks
  6. Continuous-time BRWs and aging BRWS
  7. Critical parameters for modified BRWs and GMBRWs
  8. Critical parameters for GMBRWs under finite modifications
  9. Classic critical parameters for GMBRWs and continuous-time BRWs under finite modifications
  10. An example: the tree $\mathbb{T}_d \oplus \mathbb{T}_q$

Key Result

Theorem 2.3

For every BRW $(X, {\bm{\mu}})$ and $A,B \subseteq X$, the following statements are equivalent:

Theorems & Definitions (36)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: cf:BBHZ
  • Corollary 2.4: cf:BBHZ
  • Remark 2.5
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Theorem 4.1
  • proof
  • ...and 26 more