Branching Processes: Overview

Overview| History| Surveys| Generalizations| Specializations| Related subjects| Applications| Bibliography|
The Galton-Watson process, deriving from Galton's study of extinction of family names, is a discrete-generation process parametrized by a probability distribution $p:= (p_i, i =0,1,2,\ldots)$. Each individual in a generation has a random number of offspring in the next generation, this number being picked from $p$, independently for different parents. Three cases are distinguished as follows, according to the mean number of offspring, $\mu:= \sum_i p_i$

• the subcritical case: $\mu <1$
• the critical case: $\mu =1$
• the supercritical case: $\mu >1$.

A highlight of elementary theory is that extinction is certain in the subcritical and critical cases, whereas survival forever has positive probability in the supercritical case. Branching processes, together with queues, provide the simplest examples of the criticality or phase transition phenomenon that is of widespread interest in more complex statistical physics models. A highlight of classical theory is the Kesten-Stigum theorem, asserting that in the supercritical case, where the size $X_n$ of the $n$'th generation satisfies $EX_n = \mu^n$ for mean offspring $\mu >1$, we have a non-degenerate a.s. limit for $X_n/\mu^n$, under a $L \log L$ integrability condition.

In a multitype branching process, the individuals have a type which determines the distribution of number of types of offspring. With a finite type space, the theory of multitype branching processes broadly parallels the single-type case.

Continuous-time branching processes can be defined in several levels of generality. Envisaging biological populations, one may suppose that an female has a lifetime and a number of female children born at different ages; there is an arbitrary distribution on (lifetime, ages at births), the key assumption being that for each new child there is an independent pick from the distribution. This Crump-Mode-Jagers model and its multitype analogs, in the supercritical case, retain essential features of the Galton-Watson case: there is (given nonextinction) exponential growth of population, with proportions of different type converging to a deterministic stable type structure.

The essence of classical models is the independence property of the behavior of new individuals. and lack of spatial structure. Other models incorporate further structure, but the classical theory continues to plays an important role in their study, for instance by enabling simple comparison arguments. Such models inlude

• branching random walk in which particles having spatial position,
• interacting particle systems governed by various rules for interactions between particles,
• random graphs where the structure of the component containing a particular vertex can be analyzed by comparison with a branching process.

A huge number of other variants have been studied, and around 100 papers per year continue to be published. For instance, the offspring distribution may be population-size dependent, or randomly varying in time. Classical connections with other topics such as queueing (where arrivals during the service period of one customer may be identified as offspring of that customer) and random walk (where properties of near-critical branching processes may be deduced from properties of an associated near-critical random walk) continue to be of interest. Other connections include fragmentation processes, in which particles of size $x$ are randomly split into smallers particles of masses summing to $x$, and random recursive fractals, which replace deterministic constructions of sets like the Sierpinski gasket with randomized constructions. Although elementary treatments identify the Galton-Watson process with its generation-size process $(X_n, n = 0,1,2,\ldots)$, the process is really a particular model of a random tree, and branching processes often underlie apparently different models of random tree.