Interacting Fisher–Wright diffusions in a catalytic medium

We study the longtime behaviour of interacting systems in a randomly fluctuating (space–time) medium and focus on models from population genetics. There are two prototypes of spatial models in population genetics: spatial branching processes and interacting Fisher–Wright diffusions. Quite a bit is known on spatial branching processes where the local branching rate is proportional to a random environment (catalytic medium). Here we introduce a model of interacting Fisher–Wright diffusions where the local resampling rate (or genetic drift) is proportional to a catalytic medium. For a particular choice of the medium, we investigate the longtime behaviour in the case of nearest neighbour migration on the d-dimensional lattice. While in classical homogeneous systems the longtime behaviour exhibits a dichotomy along the transience/recurrence properties of the migration, now a more complicated behaviour arises. It turns out that resampling models in catalytic media show phenomena that are new even compared with branching in catalytic medium. Received: 15 November 1999 / Revised version: 16 June 2000 / Published online: 6 April 2001     

Convergence to equilibrium of critical branching particle systems and superprocesses,and related nonlinear partial differential equations

We consider a class of multitype particle systems in ^d undergoing spatial diffusion and critical stable multitype branching, and their limits known as critical stable multitype Dawson-Watanabe processes, or superprocesses. We show that for large classes of initial states, the particle process and the superprocess converge in distribution towards known equilibrium states as time tends to infinity. As an application we obtain the asymptotic behavior of a system of nonlinear partial differential equations whose solution is related to the distribution of both the particle process and the superprocess.Research partially supported by CONACyT (Mexico), CNRS (France) and BMfWuF (Austria).     

A limit theorem for trees of alleles in branching processes with rare neutral mutations

We are interested in the genealogical structure of alleles for a Bienaymé–Galton–Watson branching process with neutral mutations (infinite alleles model), in the situation where the initial population is large and the mutation rate small. We shall establish that for an appropriate regime, the process of the sizes of the allelic sub-families converges in distribution to a certain continuous state branching process (i.e. a Ji?ina process) in discrete time. Itô’s excursion theory and the Lévy–Itô decomposition of subordinators provide fundamental insights for the results.     

Fluctuation limits of site-dependent branching systems in critical and large dimensions

In this paper, some limit processes of occupation time fluctuations of the branching particle systems with varied branching laws from site to site are obtained. The results show that the varied branching laws can not affect the limit processes and the scaling parameters in the case of large dimensions, but in the case of critical dimensions, under suitable assumptions, it changes the limit processes with simple and isotropic spatial structures to those with complicated and anisotropic spatial structures and gives log corrections in the scaling parameters.     

Backbone decomposition for continuous-state branching processes with immigration

In the spirit of Duquesne and Winkel (2007) and Berestycki et al. (2011), we show that supercritical continuous-state branching process with a general branching mechanism and general immigration mechanism is equivalent in law to a continuous-time Galton-Watson process with immigration (with Poissonian dressing). The result also helps to characterise the limiting backbone decomposition which is predictable from the work on consistent growth of Galton-Watson trees with immigration in Cao and Winkel (2010).     

Estimation of the offspring mean in a controlled branching process with a random control function

Controlled branching processes (CBP) with a random control function provide a useful way to model generation sizes in population dynamics studies, where control on the growth of the population size is necessary at each generation. An important special case of this process is the well known branching process with immigration. Motivated by the work of Wei and Winnicki [C.Z. Wei, J. Winnicki, Estimation of the mean in the branching process with immigration, Ann. Statist. 18 (1990) 1757–1773], we develop a weighted conditional least squares estimator of the offspring mean of the CBP and derive the asymptotic limit distribution of the estimator when the process is subcritical, critical and supercritical. Moreover, we show the strong consistency of this estimator in all the cases. The results obtained here extend those of Wei and Winnicki [C.Z. Wei, J. Winnicki, Estimation of the mean in the branching process with immigration, Ann. Statist. 18 (1990) 1757–1773] for branching processes with immigration and provide a unified limit theory of estimation.     

Measure-valued branching diffusions with spatial interactions

Summary A strong equation driven by a historical Brownian motion is used to construct and characterize measure-valued branching diffusions in which the spatial motions obey an Itô equation with drift and diffusion depending on the position of an individual and the entire population.     

Stochastic equations of super-Lévy processes with general branching mechanism

In this work, the process of distribution functions of a one-dimensional super-Lévy process with general branching mechanism is characterized as the pathwise unique solution of a stochastic integral equation driven by time–space Gaussian white noises and Poisson random measures. This generalizes the recent work of Xiong (2013), where the result for a super-Brownian motion with binary branching mechanism was obtained.     

Extinction properties of multi-type continuous-state branching processes

Recently in Barczy et al. (2015), the notion of a multi-type continuous-state branching process (with immigration) having $d$-types was introduced as a solution to an $d$-dimensional vector-valued SDE. Preceding that, work on affine processes, originally motivated by mathematical finance, in Duffie et al. (2003) also showed the existence of such processes. See also more recent contributions in this direction due to Gabrielli and Teichmann (2014) and Caballero and Pérez Garmendia (2017). Older work on multi-type continuous-state branching processes is more sparse but includes Watanabe (1969) and Ma (2013), where only two types are considered. In this paper we take a completely different approach and consider multi-type continuous-state branching process, now allowing for up to a countable infinity of types, defined instead as a super Markov chain with both local and non-local branching mechanisms. In the spirit of Engländer and Kypriano (2004) we explore their extinction properties and pose a number of open problems.     

The mass of super-Brownian motion upon exiting balls and Sheu’s compact support condition

We study the mass of a d$d$-dimensional super-Brownian motion as it first exits an increasing sequence of balls. The mass process is a time-inhomogeneous continuous-state branching process, where the increasing radii of the balls are taken as the time-parameter. We characterise its time-dependent branching mechanism and show that it converges, as time goes to infinity, towards the branching mechanism of the mass of a one-dimensional super-Brownian motion as it first crosses above an increasing sequence of levels.     

Parametric estimation for a simple branching diffusion process

A simple branching diffusion process is given as an elementary model of spatial evolution. A parametric estimation theory is presented for this model. As side results, a spatial central limit theorem and spatial strong law of large numbers are also obtained.     

Brownian motion on a homogeneous random fractal

Summary We introduce a simple random fractal based on the Sierpinski gasket and construct a Brownian motion upon the fractal. The properties of the process on the Sierpinski gasket are modified by the random environment. A sample path construction of the process via time truncation is used, which is a direct construction of the process on the fractal from the associated Dirichlet forms. We obtain estimates on the resolvent and transition density for the process and hence a value for the spectral dimension which satisfiesd _s=2d _f/d_w. A branching process in a random environment can be used to deduce some of the sample path properties of the process.     

Asymptotic results for exponential functionals of Lévy processes

The asymptotic behavior of expectations of some exponential functionals of a Lévy process is studied. The key point is the observation that the asymptotics only depend on the sample paths with slowly decreasing local infimum. We give not only the convergence rate but also the expression of the limiting coefficient. The latter is given in terms of some transformations of the Lévy process based on its renewal function. As an application, we give an exact evaluation of the decay rate of the survival probability of a continuous-state branching process in random environment with stable branching mechanism.     

Limit theorems for sums determined by branching and other exponentially growing processes

A branching process counted by a random characteristic has been defined as a process which at time t is the superposition of individual stochastic processes evaluated at the actual ages of the individuals of a branching population. Now characteristics which may depend not only on age but also on absolute time are considered. For supercritical processes a distributional limit theorem is proved, which implies that classical limit theorems for sums of characteristics evaluated at a fixed age point transfer into limit theorems for branching processes counted by these characteristics. A point is that, though characteristics of different individuals should be independent, the characteristics of an individual may well interplay with the reproduction of the latter. The result requires a sort of L^p-continuity for some 1 ? p ? 2. Its proof turns out to be valid for a wider class of processes than branching ones.For the case p = 1 a number of Poisson type limits follow and for p = 2 some normality approximations are concluded. For example results are obtained for processes for rare events, the age of the oldest individual, and the error of population predictions.This work has been supported by a grant from the Swedish Natural Science Research Council.     

Rate of convergence in the law of large numbers for supercritical general multi-type branching processes

We provide sufficient conditions for polynomial rate of convergence in the weak law of large numbers for supercritical general indecomposable multi-type branching processes. The main result is derived by investigating the embedded single-type process composed of all individuals having the same type as the ancestor. As an important intermediate step, we determine the (exact) polynomial rate of convergence of Nerman’s martingale in continuous time to its limit. The techniques used also allow us to give streamlined proofs of the weak and strong laws of large numbers and ratio convergence for the processes in focus.     

Weighted branching and a pathwise renewal equation

This paper is devoted to the study of a pathwise renewal equation for stochastic processes which are functions of a weighted tree defined in a general weighted branching model. Motivated by applications in the analysis of certain stochastic fixed-point equations and in the theory of general (Crump–Mode–Jagers) branching processes, we analyze the solutions to the equation under several conditions, the main result being a characterization of the set of solutions satisfying appropriate integrability conditions.     

The Spread of Branching Diffusion under Stabilizing Drift

A one-dimensional branching diffusion with a stabilizing drift is considered. We prove a theorem on the in-probability asymptotic behaviour of R ^t , the right frontier of the process over a time interval [0, t].     

Diffusion approximations of age-and-position dependent branching processes

By a (G, F, h) age-and-position dependent branching process we mean a process in which individuals reproduce according to an age dependent branching process with age distribution function G(t) and offspring distribution generating function F, the individuals (located in R^N) can not move and the distance of a new individual from its parent is governed by a probability density function h(r). For each positive integer n, let Z_n(t,dx) be the number of individuals in dx at time t of the (G, F_n,h_n) age-and-position dependent branching process. It is shown that under appropriate conditions on G, F_n and h_n, the finite dimensional distribution of $\text{Z}{}_{\mathrm{n}}\text{(nt, dx)}\text{n}$ converges, as n → ∞, to the corresponding law of a diffusion continuous state branching process X(t,dx) determined by a ψ-semigroup {ψ_t: t ? 0}. The ψ-semigroup {ψ_t} is the solution of a non-linear evolution equation. A semigroup convergence theorem due to Kurtz [10], which gives conditions for convergence in distribution of a sequence of non-Markovian processes to a Markov process, provides the main tools.