We call a random point measure infinitely ramified if for every $n\in \mathbb N$, it has the same distribution as the $n$-th generation of some branching random walk. On the other hand, branching Lévy processes model the evolution of a population in continuous time, such that individuals move in space independently, according to some Lévy process, and further beget progenies according to some Poissonian dynamics, possibly on an everywhere dense set of times. Our main result connects these two classes of processes much in the same way as in the case of infinitely divisible distributions and Lévy processes: the value at time $1$ of a branching Lévy process is an infinitely ramified point measure, and conversely, any infinitely ramified point measure can be obtained as the value at time $1$ of some branching Lévy process.
We consider an indecomposable Galton-Watson branching process with countably infinitely many types. Assuming that the process is critical and allowing for infinite variance of the offspring sizes of some (or all) types of particles we describe the asymptotic behavior of the survival probability of the process and establish a Yaglom-type conditional limit theorem for the infinite-dimensional vector of the number of particles of all types.
Consider the infinite rate mutually catalytic branching process (IMUB) constructed in [Infinite rate mutually catalytic branching in infinitely many colonies. Construction, characterization and convergence (2008) Preprint] and [Ann. Probab. 38 (2010) 479-497]. For finite initial conditions, we show that only one type survives in the long run if the interaction kernel is recurrent. On the other hand, under a slightly stronger condition than transience, we show that both types can coexist.
We construct a mutually catalytic branching process on a countable site space with infinite "branching rate". The finite rate mutually catalytic model, in which the rate of branching of one population at a site is proportional to the mass of the other population at that site, was introduced by Dawson and Perkins in [DP98]. We show that our model is the limit for a class of models and in particular for the Dawson-Perkins model as the rate of branching goes to infinity. Our process is characterized as the unique solution to a martingale problem. We also give a characterization of the process as …
We present two iterative methods for computing the global and partial extinction probability vectors for Galton-Watson processes with countably infinitely many types. The probabilistic interpretation of these methods involves truncated Galton-Watson processes with finite sets of types and modified progeny generating functions. In addition, we discuss the connection of the convergence norm of the mean progeny matrix with extinction criteria. Finally, we give a sufficient condition for a population to become extinct almost surely even though its population size explodes on the average, which is …
In the paper the rescaled occupation time fluctuation process of a certain empirical system is investigated. The system consists of particles evolving independently according to α-stable motion in R^d, α<d<2α. The particles split according to the binary critical branching law with intensity V>0. We study how the limit behaviour of the fluctuations of the occupation time depends on the \emph{initial particle configuration}. We obtain a functional central limit theorem for a vast class of infinitely divisible distributions. Our findings extend and put in a unified setting results which previousl…
We consider the extinction events of Galton-Watson processes with countably infinitely many types. In particular, we construct truncated and augmented Galton-Watson processes with finite but increasing sets of types. A pathwise approach is then used to show that, under some sufficient conditions, the corresponding sequence of extinction probability vectors converges to the global extinction probability vector of the Galton-Watson processes with countably infinitely many types. This gives rise to a number of iterative methods for the computation of the global extinction probability vector.
We prove a complete class theorem that characterizes \emph{all} stationary time reversible Markov processes whose finite dimensional marginal distributions (of all orders) are infinitely divisible. Aside from two degenerate cases (iid and constant), in both discrete and continuous time every such process with full support is a branching process with Poisson or Negative Binomial marginal univariate distributions and a specific bivariate distribution at pairs of times. As a corollary, we prove that every nondegenerate stationary integer valued processes constructed by the Markov thinning process…
In this paper, we studied the functional ergodic limits of the site-dependent branching Brownian motions in R. The results show that the limiting processes are non-degenerate if and only if the variance functions of branching laws are integrable. When the functions are integrable, although the limiting processes will vary according to the integrals, they are always positive, infinitely divisible and self-similar, and their marginal distributions are determined by a kind of 1/2-fractional integral equations. As a byproduct, the unique non-negative solutions of the integral equations are explici…
The purpose of this article is to observe that the zero sets of continuous-state branching processes with immigration (CBI) are infinitely divisible regenerative sets. Indeed, they can be constructed by the procedure of random cutouts introduced by Mandelbrot in 1972. We then show how very precise information about the zero sets of CBI can be obtained in terms of the branching and immigrating mechanism.
We introduce an algorithm for generating a random sequence of fragmentation trees, which we call the ancestral branching algorithm. This algorithm builds on the recursive partitioning structure of a tree and gives rise to an associated family of Markovian transition kernels whose finite-dimensional transition probabilities can be written in closed-form as the product over partition-valued Markov kernels. The associated tree-valued Markov process is infinitely exchangeable provided its associated partition-valued kernel is infinitely exchangeable. We also identify a transition procedure on part…
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 oth…
Branching processes are widely used to model the viral epidemic evolution. For more adequate investigation of viral epidemic modelling, we suggest to apply branching processes with transport of particles usually called branching random walks (BRWs). This allows to investigate not only the number of particles (infected individuals), but also their spatial spread. We consider two models of continuous-time BRWs on a multidimensional lattice in which the transport of infected individuals is described by a symmetric random walk on a multidimensional lattice whereas the processes of birth and death …
We study how the recurrence and transience of space-time sets for a branching random walk on a graph depends on the offspring distribution. Here, we say that a space-time set $A$ is recurrent if it is visited infinitely often almost surely on the event that the branching random walk survives forever, and say that $A$ is transient if it is visited at most finitely often almost surely. We prove that if $μ$ and $ν$ are supercritical offspring distributions with means $\bar μ< \bar ν$ then every space-time set that is recurrent with respect to the offspring distribution $μ$ is also recurrent with …
In this paper we initiate the theory of Crump-Mode-Jagers branching processes (BP) in the setting where no Malthusian parameter exist, i.e., the process grows faster than exponential. A Crump-Mode-Jagers BP is a branching process (in continuous time) where arbitrary dependencies are allowed between the birth-times of the children of a single individual in the population. It is however assumed that these reproduction processes are i.i.d. point processes for different individuals. This paper focuses on determining whether this branching process explodes, that is, the process reaches infinitely m…
We consider a general branching population where the lifetimes of individuals are i.i.d.\ with arbitrary distribution and where each individual gives birth to new individuals at Poisson times independently from each other. In addition, we suppose that individuals experience mutations at Poissonian rate $θ$ under the infinitely many alleles assumption assuming that types are transmitted from parents to offspring. This mechanism leads to a partition of the population by type, called the allelic partition. The main object of this work is the frequency spectrum $A(k,t)$ which counts the number of …
The infinitely-many-neutral-alleles model has recently been extended to a class of diffusion processes associated with Gibbs partitions of two-parameter Poisson-Dirichlet type. This paper introduces a family of infinite-dimensional diffusions associated with a different subclass of Gibbs partitions, induced by normalized inverse-Gaussian random probability measures. Such diffusions describe the evolution of the frequencies of infinitely-many types together with the dynamics of the time-varying mutation rate, which is driven by an alpha-diversity diffusion. Constructed as a dynamic version, rel…
