Related papers: Likely path to extinction for simple branching mod…

We study the asymptotic behavior of small deviation probabilities for the critical Galton-Watson processes with infinite variance of the offspring sizes of particles and apply the obtained result to investigate the structure of a reduced…

Under mild non-degeneracy assumptions on branching rates in each generation, we provide a criterion for almost-sure extinction of a multi-type branching process with time-dependent branching rates. We also provide a criterion for the total…

This paper provides a detailed analysis of the lower deviation probability properties for a $d$-type ($d>1$) Galton--Watson (GW) process $\{\textbf{Z}_n=(Z_n^{(i)})_{1\le i\le d};n\ge0\}$ in both Schr\"{o}der and B\"{o}ttcher cases. We…

The members of Martin-L\"of random closed sets under a distribution studied by Barmpalias et al. are exactly the infinite paths through Martin-L\"of random Galton--Watson trees with survival parameter $\frac{2}{3}$. To be such a member, a…

In this paper, we establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold…

Under the assumption that the initial population size of a Galton-Watson branching process increases to infinity, the paper studies asymptotic behavior of the population size before extinction. More specifically, we establish asymptotic…

We consider the simple random walk on Galton-Watson trees with supercritical offspring distribution, conditioned on non-extinction. In case the offspring distribution has finite support, we prove an upper bound for the annealed return…

Branching processes $(Z_n)_{n \ge 0}$ in a varying environment generalize the Galton-Watson process, in that they allow time-dependence of the offspring distribution. Our main results concern general criteria for a.s. extinction,…

We prove a sample path Large Deviation Principle (LDP) for a class of jump processes whose rates are not uniformly Lipschitz continuous in phase space. Building on it we further establish the corresponding Wentzell-Freidlin (W-F) (infinite…

Extinction appears ubiquitously in many fields, including chemical reactions, population biology, evolution, and epidemiology. Even though extinction as a random process is a rare event, its occurrence is observed in large finite…

We develop a theory of first passage processes in stochastic non-equilibrium systems of birth-death type using two closely related epidemiological models as examples. Our method employs the probability generating function technique in…

Branching processes pervade many models in statistical physics. We investigate the survival probability of a Galton-Watson branching process after a finite number of generations. We reveal the finite-size scaling law of the survival…

The standard Large Deviation Theory (LDT) is mathematically illustrated by the Boltzmann-Gibbs factor which describes the thermal equilibrium of short-range-interacting many-body Hamiltonian systems, the velocity distribution of which is…

Recent progress in the study of the contact process [2] has verified that the extinction-survival threshold $\lambda_1$ on a Galton-Watson tree is strictly positive if and only if the offspring distribution $\xi$ has an exponential tail. In…

In this article we prove a local large deviation principle (LLDP) for the critical multitype Galton-Watson process from spectral potential point. We define the so-called a spectral potential $U_{\skrik}(\,\cdot,\,\pi)$ for the Galton-Watson…

We establish large deviation principle (LDP) for the family of vector-valued random processes $(X^\epsilon,Y^\epsilon),\epsilon\to 0$ defined as $$ X^\epsilon_t=\frac{1}{\epsilon^\kappa}\int_0^t H(\xi^\epsilon_s,Y^\epsilon_s)ds,…

Extinction of an epidemic or a species is a rare event that occurs due to a large, rare stochastic fluctuation. Although the extinction process is dynamically unstable, it follows an optimal path that maximizes the probability of…

In this note we give a new method for getting a series of approximations for the extinction probability of the one-dimensional contact process by using the Gr\"obner basis.

Conditions for almost sure extinction are studied in discrete time branching processes with an infinite number of types. It is not assumed that the expected number of children is a bounded function of the parent's type. There might also be…

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…