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).     

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     

Stochastic equations of non-negative processes with jumps

We study stochastic equations of non-negative processes with jumps. The existence and uniqueness of strong solutions are established under Lipschitz and non-Lipschitz conditions. Under suitable conditions, the comparison properties of solutions are proved. Those results are applied to construct continuous state branching processes with immigration as strong solutions of stochastic equations.     

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.     

Blowup for the heat equation with a noise term

Summary In this paper we study blow up of the equation , where is a two-dimensional white noise field and where Dirichlet boundary conditions are enforced. It is known that if <3/2, then the solution exists for all time; in this paper we show that if is much larger than 3/2, then the solution blows up in finite time with positive probability. We prove this by considering how peaks in the solution propagate. If a peak of high mass forms, we rescale the equation and divide the mass of the peak into a collection of peaks of smaller mass, and these peaks evolve almost independently. In this way we compare the evolution ofu to a branching process. Large peaks are regarded as particles in this branching process. Offspring are peaks which are higher by some factor. We show that the expected number of offspring is greater than one when is much larger than 3/2, and thus the branching process survives with positive probability, corresponding to blowup in finite time.Supported by NSF grant DMS-9021508, NSA grant MDA904-910-H-0034, and ARO Grant MSI DAAL03-91-C-0027Supported by ONR grant N00014-91-J-1526.     

A continuous-state polynomial branching process

A continuous-state polynomial branching process is constructed as the pathwise unique solution of a stochastic integral equation with absorbing boundary condition. The process can also be obtained from a spectrally positive Lévy process through Lamperti type transformations. The extinction and explosion probabilities and the mean extinction and explosion times are computed explicitly. Some of those are also new for the classical linear branching process. We present necessary and sufficient conditions for the process to extinguish or explode in finite times. In the critical or subcritical case, we give a construction of the process coming down from infinity. Finally, it is shown that the continuous-state polynomial branching process arises naturally as the rescaled limit of a sequence of discrete-state processes.     

Stochastic p.d.e.'s arising from the long range contact and long range voter processes

Summary A long range contact process and a long range voter process are scaled so that the distance between sites decreases and the number of neighbors of each site increases. The approximate densities of occupied sites, under suitable tine scaling, converge to continuous space time densities which solve stochastic p.d.e.'s. For the contact process the limiting equation is the Kolmogorov-Petrovskii-Piscuinov equation driven by branching white noise. For the voter process the limiting equation is the heat equation driven by Fisher-Wright white noise.     

FBSDEs with time delayed generators: L-solutions, differentiability, representation formulas and path regularity

We extend the work of Delong and Imkeller (2010) [6] and [7] concerning backward stochastic differential equations with time delayed generators (delay BSDEs). We give moment and a priori estimates in general L^p-spaces and provide sufficient conditions for the solution of a delay BSDE to exist in L^p. We introduce decoupled systems of SDEs and delay BSDEs (delay FBSDEs) and give sufficient conditions for their variational differentiability. We connect these variational derivatives to the Malliavin derivatives of delay FBSDEs via the usual representation formulas. We conclude with several path regularity results, in particular we extend the classic L²-path regularity to delay FBSDEs.     

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.