arXiv subjects

Natalia Cardona-Tobón

Publications and source records attributed to Natalia Cardona-Tobón.

10 recordsLinked to original sources

arXiv2026

The contact process with fitness on random trees

The contact process is a simple model for the spread of an infection in a structured population. We consider a variant of this process on Bienaymé-Galton-Watson trees, where vertices are equipped with a random fitness representing inhomogeneous transmission rates among individuals. In this paper, we establish conditions under which this inhomogeneous contact process exhibits a phase transition. We first prove that if certain mixed moments of the joint offspring and fitness distribution are finite, then the survival threshold is strictly positive. Further, we show that, if slightly different mixed moments are infinite, then this implies that there is no phase transition and the process survives with positive probability for any choice of the infection parameter. A similar dichotomy is known for the contact process on a Bienaymé-Galton-Watson tree. However, we show that the introduction of fitness means that we have to take into account the combined effect of fitness and offspring distribution to decide which scenario occurs.

math.PR↗

arXiv2026

Critical non-local spatial branching processes with infinite variance conditioned on survival

We consider the setting of either a general non-local branching particle process or a general non-local superprocess. Under the assumption that the mean semigroup has a Perron-Frobenious type behaviour in combination with a regularly varying assumption on the reproductive point process, which permits infinite second moments, we consider sufficient conditions that ensure limiting distributional stability when conditioned on survival at criticality. We offer two main results. Under the aforesaid conditions, our first main contribution establishes the polynomial decay in time of the survival probability in the spirit of a classical Kolmogorov limit. The second main contribution pertains to the stability, when conditioning on survival, in the spirit of a Yaglom limit. In both cases our proofs work equally well for the analogous setting of superprocesses with non-local branching. Our results complete a series of articles for various families of non spatial branching processes from the 1950s-1970s as well complementing a recent result of this type for spatial branching processes of Ren, Song and Sun (2020). The generality of our results improve on older work of Hering and Hoppe (1981) and Asmussen and Hering (1983) who dealt with a similar context for branching particle systems, as well as providing a general framework for non-local superprocesses.

math.PR↗

arXiv2026

The contact process on dynamical random trees with degree dependence

The contact process is a simple model for the spread of an infection in a structured population. We investigate the case when the underlying structure evolves dynamically as a degree-dependent dynamical percolation model. Starting with a connected locally finite base graph we initially declare edges independently open with a connection probability that is allowed to depend on the degree of the adjacent vertices and closed otherwise. Edges are independently updated with a rate depending on the degrees and then are again declared open and closed with the same probabilities. We are interested in the contact process, where infections are only allowed to spread via open edges. Our aim is to analyse the impact of the update speed and the connection probability on the existence of a phase transition. For a general connected locally finite graph, our first result gives sufficient conditions for the critical value for survival to be strictly positive. Furthermore, in the setting of Bienaymé-Galton-Watson trees, we show that the process survives strongly with positive probability for any infection rate if the offspring distribution has a stretched exponential tail with an exponent depending on the connection probability and the update speed. In particular, if the offspring distribution follows a power law and the connection probability is given by a product kernel and the update speed exhibits polynomial behaviour, we provide a complete characterisation of the phase transition.

math.PR↗

arXiv2025

The coalescent of a sample from a linear-fractional branching process

In this article, we focus on Bienaymé-Galton-Watson processes with linear-fractional offspring distributions. At a fixed generation, we consider a sample of the individuals alive, drawn in two different ways: either through Bernoulli sampling, where each individual is selected independently with a given probability, or through uniform sampling, where a fixed number of individuals are chosen uniformly at random. We analyze the genealogical trees generated by the sampled individuals. In particular, we establish a relationship between the distributions of the trees resulting from the two sampling schemes.

math.PR↗

arXiv2024

Rates on Yaglom's limit for Galton-Watson processes in a varying environment

A Galton-Watson process in a varying environment is a discrete time branching process where the offspring distributions vary among generations. It is known that in the critical case, these processes have a Yaglom limit, that is, a suitable normalization of the process conditioned on non-extinction converges in distribution to a standard exponential random variable. In this manuscript, we provide the rate of convergence of the Yaglom limit with respect to the Wasserstein metric.

math.PR↗

arXiv2024

A min-max random game on a graph that is not a tree

We study a random game in which two players in turn play a fixed number of moves. For each move, there are two possible choices. To each possible outcome of the game we assign a winner in an i.i.d. fashion with a fixed parameter p. In the case where all different game histories lead to different outcomes, a classical result due to Pearl (1980) says that in the limit when the number of moves is large, there is a sharp threshold in the parameter p that separates the regimes in which either player has with high probability a winning strategy. We are interested in a modification of this game where the outcome is determined by the exact sequence of moves played by the first player and by the number of times the second player has played each of the two possible moves. We show that also in this case, there is a sharp threshold in the parameter p that separates the regimes in which either player has with high probability a winning strategy. Since in the modified game, different game histories can lead to the same outcome, the graph associated with the game is no longer a tree which means independence is lost. As a result, the analysis becomes more complicated and open problems remain.

math.PR↗

arXiv2024

Explosion rates for continuous-state branching processes in a Lévy environment

Here, we study the long-term behaviour of the non-explosion probability for continuous-state branching processes in a Lévy environment when the branching mechanism is given by the negative of the Laplace exponent of a subordinator. In order to do so, we study the law of this family of processes in the infinite mean case and provide necessary and sufficient conditions for the process to be conservative, i.e. that the process does not explode in finite time a.s. In addition, we establish precise rates for the non-explosion probabilities in the subcritical and critical regimes, first found by Palau et al. [19] in the case when the branching mechanism is given by the negative of the Laplace exponent of a stable subordinator.

math.PR↗

arXiv2023

Speed of extinction for continuous state branching processes in subcritical Lévy environments: the strongly and intermediate regimes

In this paper, we study the speed of extinction of continuous state branching processes in subcritical Lévy environments. More precisely, when the associated Lévy process to the environment drifts to $-\infty$ and, under a suitable exponential martingale change of measure (Esscher transform), the environment either drifts to $-\infty$ or oscillates. We extend recent results of Palau et al. (2016) and Li and Xu (2018), where the branching term is associated to a spectrally positive stable Lévy process and complement the recent article of Bansaye et al. (2021) where the critical case was studied. Our methodology combines a path analysis of the branching process together with its Lévy environment, fluctuation theory for Lévy processes and the asymptotic behaviour of exponential functionals of Lévy processes. As an application of the aforementioned results, we characterise the process conditioned to survival also known as the $Q$-process.

math.PR↗

arXiv2023

Speed of extinction for continuous state branching processes in a weakly subcritical Lévy environment

In this manuscript, we continue with the systematic study of the speed of extinction of continuous state branching processes in Lévy environments under more general branching mechanisms. Here, we deal with the weakly subcritical regime under the assumption that the branching mechanism is regularly varying. We extend recent results of Li and Xu [14] and Palau et al. [17], where it is assumed that the branching mechanism is stable and complement the recent articles of Bansaye et al. [2] and by the authors in [7], where the critical and the strongly and intermediate subcritical cases were treated, respectively. Our methodology combines a path analysis of the branching process together with its Lévy environment, fluctuation theory for Lévy processes and the asymptotic behaviour of exponential functionals of Lévy processes. Our approach is inspired by Afanasyev et al. [1], where the discrete analogue was obtained, and by [2] and [7].

math.PR↗

arXiv2020

Yaglom's limit for critical Galton-Watson processes in varying environment: A probabilistic approach

A Galton-Watson process in varying environment is a discrete time branching process where the offspring distributions vary among generations. Based on a two-spine decomposition technique, we provide a probabilistic argument of a Yaglom-type limit for this family processes. The result states that, in the critical case, a suitable normalisation of the process conditioned on non-extinction converges in distribution to an exponential random variable. Recently, this result has been established by Kersting [{\it J. Appl. Probab.} {\bf57}(1), 196--220, 2020] using analytic techniques.

math.PR↗