arXiv subjects

Daniela Bertacchi

Publications and source records attributed to Daniela Bertacchi.

At least 19 recordsLinked to original sources

arXiv2026

Branching random walks with ageing

Branching processes are models used to describe populations that reproduce and die over time. In the classical setting, an individual's reproductive capacity remains constant throughout its lifetime. However, in real-world situations, reproductive capacity typically undergoes ageing - that is, after reaching a peak, it decreases over time. In this work, we study the influence of ageing on the behaviour of the process and how modifying its parameters, along with reproduction rates, affects the destiny of the process.

math.PR↗

arXiv2026

Critical parameters of germ-monotone families of branching random walks

We introduce a broad class of families of branching random walks on a countable set $X$, which we refer to as germ-monotone branching random walks (GMBRWs). The processes in each family are parametrized by a positive parameter $\lambda>0$, which controls the overall reproductive speed, and they are monotonically increasing in $\lambda$ with respect to the germ order, a notion that extends classical stochastic domination. This framework encompasses a wide range of models, including classical continuous-time branching random walks, as well as discrete-time counterparts of certain non-Markovian processes such as ageing branching random walks. We define a general notion of critical parameter $\lambda(A)$ associated with each subset $A \subseteq X$, which serves as a threshold separating almost sure extinction in $A$ from positive probability of survival in $A$. This unifies and extends the classical global and local critical parameters $\lambda_w$ and $\lambda_s$, which can be recovered as special cases. We then investigate how modifications of the reproduction laws, either on a finite set or on a more general subset of $X$, affect these critical parameters. Our results extend earlier contributions in the literature.

math.PR↗

arXiv2025

On the critical parameters of branching random walks

Given a discrete spatial structure $X$, we define continuous-time branching processes that model a population breeding and dying on $X$. These processes are usually called branching random walks. They are characterized by breeding rates $k_{xy}$ (governing the rate at which individuals at $x$ send offspring to $y$), and by a multiplicative speed parameter $\lambda$. These processes also serve as models for epidemic spreading, where $\lambda k_{xy}$ represents the infection rate from $x$ to $y$. Two critical parameters of interest are the global critical parameter $\lambda_w$, related to global survival, and the local critical parameter $\lambda_s$, related to survival within finite sets (with $\lambda_w\le\lambda_s$). In disease control, the primary goal is to lower $\lambda$ so that the process dies out, at least locally. Nevertheless, a process that survives globally can still pose a threat, especially if sudden changes cause global survival to transition into local survival. In fact, local modifications to the rates can affect the values of both critical parameters, making it important to understand when and how they can be increased. Using results on the comparison of extinction probabilities for a single branching random walk across different sets, we extend the analysis to extinction probabilities and critical parameters of pairs of branching random walks whose rates coincide outside a fixed set $A \subseteq X$. We say that two branching random walks are equivalent if their rates coincide everywhere except on a finite subset of $X$. Given an equivalence class of branching random walks, we prove that if one process has $\lambda^*_w \neq \lambda^*_s$, then $\lambda^*_w$ is the maximal possible value of this parameter within the class. We describe the possible configurations for the critical parameters within these equivalence classes.

math.PR↗

arXiv2025

Results on branching random walks and rumor processes via germ order

Germ order is a non-standard stochastic order defined through the comparison of the generating functions of the processes. This order was first introduced for branching random walks with a constant breeding law and independent dispersal of offspring, which are characterized by a one-dimensional generating function. In this work, we investigate the properties of the extension of this concept to processes characterized by a multidimensional generating function, such as general branching random walks and rumor processes. In particular, we use germ ordering to characterize the behavior of certain branching random walks and rumor processes with inhomogeneous breeding/transmitting laws.

math.PR↗

arXiv2024

Strong survival and extinction for multitype branching processes via a new order for generating functions

We consider general discrete-time multitype branching processes on a countable set $X$. According to these processes, a particle of type $x\in X$ generates a random number of children and chooses their type in $X$, not necessarily independently nor with the same law for different parent types. We introduce a new type of stochastic ordering of multitype branching processes, generalizing the germ order introduced by Hutchcroft in arXiv:2011.06402, which relies on the generating function of the process. We prove that given two multitype branching processes with law $\mathbf{\mu}$ and $\mathbf{\nu}$ respectively, with $\mathbf{\mu}\ge\mathbf{\nu}$, then in every set where there is survival according to $\\mathbf{\nu}$, there is survival also according to $\mathbf{\mu}$. Moreover, in every set where there is strong survival according to $\mathbf{\nu}$, there is strong survival also according to $\mathbf{\mu}$, provided that the supremum of the global extinction probabilities, for the $\mathbf{\nu}$-process, taken over all starting points $x$, is strictly smaller than 1. New conditions for survival and strong survival for inhomogeneous multitype branching processes are provided. We also extend a result of Moyal which claims that, under some conditions, the global extinction probability for a multitype branching process is the only fixed point of its generating function, whose supremum over all starting coordinates may be smaller than 1.

math.PR↗

arXiv2023

Martin boundaries and asymptotic behavior of branching random walks

Let $G$ be an infinite, locally finite graph. We investigate the relation between supercritical, transient branching random walk and the Martin boundary of its underlying random walk. We show results regarding the typical asymptotic directions taken by the particles, and as a consequence we find a new connection between $t$-Martin boundaries and standard Martin boundaries. Moreover, given a subgraph $U$ we study two aspects of branching random walks on $U$: when the trajectories visit $U$ infinitely often (survival) and when they stay inside $U$ forever (persistence). We show that there are cases, when $U$ is not connected, where the branching random walk does not survive in $U$, but the random walk on $G$ converges to the boundary of $U$ with positive probability. In contrast, the branching random walk can survive in $U$ even though the random walk eventually exits $U$ almost surely. We provide several examples and counterexamples.

math.PR↗

arXiv2022

An epidemic model in inhomogeneous environment

The current work deals with an epidemic model on the complete graph K_n on n vertices in a non-homogeneous setting, where the vertices may have distinct types. Different types differ in the probability of getting infected, and/or in the capacity of infecting other vertices. This generalizes previous models where vertices are all of the same type and have equal probabilities of being infected. We prove laws of large numbers and central limit theorems for the the total duration of the process and for the number of infected vertices, respectively, when n goes to infinity. By coupling the epidemic model with a Poisson process, we also obtain continuous-time counterparts of the above-mentioned limit results. Moreover, we also prove that when all individuals have the same spread capacity, then a population with inhomogeneous susceptibility is less affected by the epidemics than a homogeneous population.

math.PR↗

arXiv2022

An evolution model with uncountably many alleles

We study a class of evolution models, where the breeding process involves an arbitrary exchangeable process, allowing for mutations to appear. The population size $n$ is fixed, hence after breeding, selection is applied. Individuals are characterized by their genome, picked inside a set $X$ (which may be uncountable), and there is a fitness associated to each genome. Being less fit implies a higher chance of being discarded in the selection process. The stationary distribution of the process can be described and studied. We are interested in the asymptotic behavior of this stationary distribution as $n$ goes to infinity. Choosing a parameter $λ>0$ to tune the scaling of the fitness when $n$ grows, we prove limiting theorems both for the case when the breeding process does not depend on $n$, and for the case when it is given by a Dirichlet process prior. In both cases, the limit exhibits phase transitions depending on the parameter $λ

math.PR↗

arXiv2020

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

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

math.PR↗

arXiv2018

Branching random walks with uncountably many extinction probability vectors

Given a branching random walk on a set $X$, we study its extinction probability vectors $\mathbf q(\cdot,A)$. Their components are the probability that the process goes extinct in a fixed $A\subseteq X$, when starting from a vertex $x\in X$. The set of extinction probability vectors (obtained letting $A$ vary among all subsets of $X$) is a subset of the set of the fixed points of the generating function of the branching random walk. In particular here we are interested in the cardinality of the set of extinction probability vectors. We prove results which allow to understand whether the probability of extinction in a set $A$ is different from the one of extinction in another set $B$. In many cases there are only two possible extinction probability vectors and so far, in more complicated examples, only a finite number of distinct extinction probability vectors had been explicitly found. Whether a branching random walk could have an infinite number of distinct extinction probability vectors was not known. We apply our results to construct examples of branching random walks with uncountably many distinct extinction probability vectors.

math.PR↗

arXiv2018

Recent results on branching random walks

This paper is a collection of recent results on discrete-time and continuous-time branching random walks. Some results are new and others are known. Many aspects of this theory are considered: local, global and strong local survival, the existence of a pure global survival phase and the approximation of branching random walks by means of multitype contact processes or spatially confined branching random walks. Most results are obtained using a generating function approach: the probabilities of extinction are seen as fixed points of an infinite dimensional power series. Throughout this paper we provide many nontrivial examples and counterexamples.

math.PR↗

arXiv2018

A stochastic model for the evolution of species with random fitness

We generalize the evolution model introduced by Guiol, Machado and Schinazi (2010). In our model at odd times a random number X of species is created. Each species is endowed with a random fitness with arbitrary distribution on $[0, 1]$. At even times a random number Y of species is removed, killing the species with lower fitness. We show that there is a critical fitness $f_c$ below which the number of species hits zero i.o. and above of which this number goes to infinity. We prove uniform convergence for the distribution of surviving species and describe the phenomena which could not be observed in previous works with uniformly distributed fitness.

math.PR↗

arXiv2017

Galton-Watson processes in varying environment and accessibility percolation

This paper deals with branching processes in varying environment, namely, whose offspring distributions depend on the generations. We provide sufficient conditions for survival or extinction which rely only on the first and second moments of the offspring distributions. These results are then applied to branching processes in varying environment with selection where every particle has a real-valued label and labels can only increase along genealogical lineages; we obtain analogous conditions for survival or extinction. These last results can be interpreted in terms of accessibility percolation on Galton-Watson trees, which represents a relevant tool for modeling the evolution of biological populations.

math.PR↗

arXiv2017

Global survival of branching random walks and tree-like branching random walks

The reproduction speed of a continuous-time branching random walk is proportional to a positive parameter $λ$. There is a threshold for $λ$, which is called $λ_w$, that separates almost sure global extinction from global survival. Analogously, there exists another threshold $λ_s$ below which any site is visited almost surely a finite number of times (i.e.~local extinction) while above it there is a positive probability of visiting every site infinitely many times. The local critical parameter $λ_s$ is completely understood and can be computed as a function of the reproduction rates. On the other hand, only for some classes of branching random walks it is known that the global critical parameter $λ_w$ is the inverse of a certain function of the reproduction rates, which we denote by $K_w$. We provide here new sufficient conditions which guarantee that the global critical parameter equals $1/K_w$. This result extends previously known results for branching random walks on multigraphs and general branching random walks. We show that these sufficient conditions are satisfied by periodic tree-like branching random walks. We also discuss the critical parameter and the critical behaviour of continuous-time branching processes in varying environment. So far, only examples where $λ_w=1/K_w$ were known; here we provide an example where $λ_w>1/K_w$.

math.PR↗

arXiv2016

A generating function approach to branching random walks

It is well known that the behaviour of a branching process is completely described by the generating function of the offspring law and its fixed points. Branching random walks are a natural generalization of branching processes: a branching process can be seen as a one-dimensional branching random walk. We define a multidimensional generating function associated to a given branching random walk. The present paper investigates the similarities and the differences of the generating functions, their fixed points and the implications on the underlying stochastic process, between the one-dimensional (branching process) and the multidimensional case (branching random walk). In particular, we show that the generating function of a branching random walk can have uncountably many fixed points and a fixed point may not be an extinction probability, even in the irreducible case (extinction probabilities are always fixed points). Moreover, the generating function might not be a convex function. We also study how the behaviour of a branching random walk is affected by local modifications of the process. As a corollary, we describe a general procedure with which we can modify a continuous-time branching random walk which has a weak phase and turn it into a continuous-time branching random walk which has strong local survival for large or small values of the parameter and non-strong local survival for intermediate values of the parameter.

math.PR↗

arXiv2015

Branching random walks and multi-type contact-processes on the percolation cluster of ${\mathbb{Z}}^{d}$

In this paper we prove that, under the assumption of quasi-transitivity, if a branching random walk on ${\mathbb{Z}^d}$ survives locally (at arbitrarily large times there are individuals alive at the origin), then so does the same process when restricted to the infinite percolation cluster ${\mathcal{C}_{\infty}}$ of a supercritical Bernoulli percolation. When no more than $k$ individuals per site are allowed, we obtain the $k$-type contact process, which can be derived from the branching random walk by killing all particles that are born at a site where already $k$ individuals are present. We prove that local survival of the branching random walk on ${\mathbb{Z}^d}$ also implies that for $k$ sufficiently large the associated $k$-type contact process survives on ${\mathcal{C}_{\infty}}$. This implies that the strong critical parameters of the branching random walk on ${\mathbb{Z}^d}$ and on ${\mathcal{C}_{\infty}}$ coincide and that their common value is the limit of the sequence of strong critical parameters of the associated $k$-type contact processes. These results are extended to a family of restrained branching random walks, that is, branching random walks where the success of the reproduction trials decreases with the size of the population in the target site.

math.PR↗

arXiv2015

The timing of life history events in presence of soft disturbances

We study a model for the evolutionarily stable strategy (ESS) used by biological populations for choosing the time of life-history events, such as migration and breeding. In our model we accounted for both intra-species competition (early individuals have a competitive advantage) and a disturbance which strikes at a random time, killing a fraction $1-p$ of the population. Disturbances include spells of bad weather, such as freezing or heavily raining days. It has been shown in Iwasa and Levin (1995), that when $p=0$, then the ESS is a mixed strategy, where individuals wait for a certain time and afterwards start arriving (or breeding) every day. We remove the constraint $p=0$ and show that if $0<p<1$ then the ESS still implies a mixed choice of times, but strong competition may lead to a massive arrival at the earliest time possible of a fraction of the population, while the rest will arrive throughout the whole period during which the disturbance may occur. More precisely, given $p$, there is a threshold for the competition parameter $a$, above which massive arrivals occur and below which there is a behaviour as in Iwasa and Levin (1995). We study the behaviour of the ESS and of the average fitness of the population, depending on the parameters involved. We also discuss how the population may be affected by climate change, in two respects: first, how the ESS should change under the new climate and whether this change implies an increase of the average fitness; second, which is the impact of the new climate on a population that still follows the old strategy. We show that, at least under some conditions, extreme weather events imply a temporary decrease of the average fitness (thus an increasing mortality). If the population adapts to the new climate, the survivors may have a larger fitness.

math.CA↗

arXiv2013

Rumor processes in random environment on N and on Galton-Watson trees

The aim of this paper is to study rumor processes in random environment. In a rumor process a signal starts from the stations of a fixed vertex (the root) and travels on a graph from vertex to vertex. We consider two rumor processes. In the firework process each station, when reached by the signal, transmits it up to a random distance. In the reverse firework process, on the other hand, stations do not send any signal but they "listen" for it up to a random distance. The first random environment that we consider is the deterministic 1-dimensional tree N with a random number of stations on each vertex; in this case the root is the origin of N. We give conditions for the survival/extinction on almost every realization of the sequence of stations. Later on, we study the processes on Galton-Watson trees with random number of stations on each vertex. We show that if the probability of survival is positive, then there is survival on almost every realization of the infinite tree such that there is at least one station at the root. We characterize the survival of the process in some cases and we give sufficient conditions for survival/extinction.

math.PR↗