A Conversation with Don Dawson

Contemporary mathematics, 2003

Stochastic Processes and their Applications, 1995

Fleming-Viot processes and Dawson-Watanabe processes are two classes of "superprocesscs" that have received a great deal of attention in recent years. These processes have many properties in common. In this paper, we prove a result that helps to explain why this is so. It allows one to prove certain theorems for one class when they are true for the other. More specifically, we show that product moments of a Fleming-Viot process can be bounded above by the corresponding moments of the Dawson-Watanabe process with the same "underlying particle motion", and vice versa except for a multiplicative constant. As an application. we establish existence and continuity properties of local time for certain Fleming-Viot processes.

2018

Many studies in Economics and other disciplines have been reporting distributions following power-law behavior (i.e distributions of incomes (Pareto’s law), city sizes (Zipf’s law), frequencies of words in long sequences of text etc.)[1, 6, 7]. This widespread observed regularity has been explained in many ways: generalized Lotka-Volterra (GLV) equations, self-organized criticality and highly optimized tolerance [2,3,4]. The evolution of the phenomena exhibiting power-law behavior is often considered to involve a varying, but size independent, proportional growth rate, which mathematically can be modeled by geometric Brownian motion (GBM) dXt = rtXtdt+αXtWt where Wt is white noise or the increment of a Wiener process. It is the primary purpose of this article to study both the upper tail and lower tail of the distribution following the geometric Brownian motion and to correlate this study with recent results showing the emergence of power-law behavior from heterogeneous interacting ...

arXiv: Probability, 2019

In this paper, we consider Galton-Watson processes with immigration. Pick $i(\ge2)$ individuals randomly without replacement from the $n$-th generation and trace their lines of descent back in time till they coalesce into $1$ individual in a certain generation, which we denote by $X_{i,1}^n$ and is called the coalescence time. Firstly, we give the probability distribution of $X_{i,1}^n$ in terms of the probability generating functions of both the offspring distribution and the immigration law. Then by studying the limit behaviors of various functionals of the Galton-Watson process with immigration, we find the limit distribution of $X_{2,1}^n$ as $n\rightarrow\infty.$

Problems of Information Transmission, 2018

arXiv (Cornell University), 2007

A multitype Dawson-Watanabe process is conditioned, in subcritical and critical cases, on non-extinction in the remote future. On every finite time interval, its distribution is absolutely continuous with respect to the law of the unconditioned process. A martingale problem characterization is also given. Several results on the long time behavior of the conditioned mass process-the conditioned multitype Feller branching diffusion-are then proved. The general case is considered first, where the mutation matrix which models the interaction between the types, is irreducible. Several two-type models with decomposable mutation matrices are also analyzed.

2017

Modern natural sciences and even more so biophysics are unthinkable without the understanding of the fundamental nature of the randomness in the time evolution of quantities of interest. The first international and interdisciplinary workshop STOCHASTIC PRO-CESSES WITH APPLICATIONS IN THE NATURAL SCIENCES held at Universidad de los Andes from December 5th to 9th in 2016 offered four lecture courses to cover a section of this enormous and prolific field. Two of these lectures were centered in biophysics and two of them in the mathematical theory of stochastic processes, ranging from probabilistic modeling of the degradation of cell molecules, biological functional circuits via an introductory mathematics lecture on branching processes and a more advanced mathematics course on the dynamics of Markov processes. The courses were aimed at Master's and Ph.D. students as well as ambitious undergraduates. The courses in physical biology were focused on the nature and modeling of random effects of particular classes of biological systems. Both lecturers are distinguished researchers in the field, Dr. Angelo Valleriani from Germany, and Prof. Juan Manuel Pedraza from the Physics department of Universidad de los Andes, Colombia. These notes include the chapter Stochastic modeling of complex macromolecular decay by Dr. Valleriani, group leader of the team Stochastic processes in complex and biological systems from the Max-Planck-Institute of Colloids and Interfaces, Potsdam. They are based on the classes he taught in Bogotá and yield beyond the specific modeling problem a general perspective on the modeling of gene expression. The first mathematics course offered a profound introduction to a class of stochastic processes which are particularly important in system biology, so-called branching processes. It was held by Prof. Sylvie Roelly, Vice-Dean of the Faculty of Science at the University of Potsdam and winner of the Itô prize in 2007 with wide expertise in stochastic processes. These lecture notes contain the chapter Algunas propriedades básicas de procesos de ramificación (in Spanish) as an introduction to the field of branching v Contents 1 Stochastic modeling of complex macromolecular decay

The Annals of Applied Probability, 2011

We study the evolution of a particle system whose genealogy is given by a supercritical continuous time Galton-Watson tree. The particles move independently according to a Markov process and when a branching event occurs, the offspring locations depend on the position of the mother and the number of offspring. We prove a law of large numbers for the empirical measure of individuals alive at time t. This relies on a probabilistic interpretation of its intensity by mean of an auxiliary process. The latter has the same generator as the Markov process along the branches plus additional jumps, associated with branching events of accelerated rate and biased distribution. This comes from the fact that choosing an individual uniformly at time t favors lineages with more branching events and larger offspring number. The central limit theorem is considered on a special case. Several examples are developed, including applications to splitting diffusions, cellular aging, branching Lévy processes.

2019

We consider the problem of estimating the elapsed time since the most recent common ancestor of a finite random sample drawn from a population which has evolved through a Bienaymé-Galton-Watson branching process. More specifically, we are interested in the diffusion limit appropriate to a supercritical process in the near-critical limit evolving over a large number of time steps. Our approach differs from earlier analyses in that we assume the only known information is the mean and variance of the number of offspring per parent, the observed total population size at the time of sampling, and the size of the sample. We obtain a formula for the probability that a finite random sample of the population is descended from a single ancestor in the initial population, and derive a confidence interval for the initial population size in terms of the final population size and the time since initiating the process. We also determine a joint likelihood surface from which confidence regions can ...

arXiv (Cornell University), 2017

We consider a class of birth-and-death processes describing a population made of d sub-populations of different types which interact with one another. The state space is Z d + (unbounded). We assume that the population goes almost surely to extinction, so that the unique stationary distribution is the Dirac measure at the origin. These processes are parametrized by a scaling parameter K which can be thought as the order of magnitude of the total size of the population at time 0. For any fixed finite time span, it is well-known that such processes, when renormalized by K, are close, in the limit K → +∞, to the solutions of a certain differential equation in R d + whose vector field is determined by the birth and death rates. We consider the case where there is a unique attractive fixed point (off the boundary of the positive orthant) for the vector field (while the origin is repulsive). What is expected is that, for K large, the process will stay in the vicinity of the fixed point for a very long time before being absorbed at the origin. To precisely describe this behavior, we prove the existence of a quasi-stationary distribution (qsd, for short). In fact, we establish a bound for the total variation distance between the process conditioned to non-extinction before time t and the qsd. This bound is exponentially small in t, for t log K. As a by-product, we obtain an estimate for the mean time to extinction in the qsd. We also quantify how close is the law of the process (not conditioned to non-extinction) either to the Dirac measure at the origin or to the qsd, for times much larger than log K and much smaller than the mean time to extinction, which is exponentially large as a function of K. Let us stress that we are interested in what happens for finite K. We obtain results much beyond what large deviation techniques could provide.