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ADVANCED BRANCHING processes: (i) branching processes in random PROCESSES environment, which are examples of branch- ing processes, where the dynamic evolves FLORIAN SIMATOS (randomly) over time; (ii) branching random Eindhoven University of walks that exhibit a spatial feature; and Technology, Eindhoven, (iii) continuous state branching processes Netherlands (CSBPs) that can be seen as continuous approximations of Galton—Watson pro- cesses where both time and space are continuous. The presentation of CSBPs INTRODUCTION will also be a good place to briefly discuss superprocesses. We each time focus on the Branching processes have their roots in the most basic properties of these processes, study of the so-called family name extinction such as the extinction probability or the problem (see Introduction to Branching behavior of extremal particles. Processes) and find their most natural and This choice of topics does not aim to be important applications in biology, especially exhaustive and reflects a personal selec- in the study of population dynamics. They tion of exciting and recent research on were also motivated by the study of nuclear branching processes. It leaves out certain fission reactions and underwent rapid devel- important classes of models, which include opment during the Manhattan project under multitype branching processes, branching the impulse of Szilard´ and Ulam. To date, processes with immigration, and population- they continue to be very important in reac- size-dependent branching processes. For tor physics. They also play a major role in multitype branching processes, the book (applied) probability at large, and appear in by Mode [1] offers a good starting point. a wide variety of problems in queuing theory, Branching processes with immigration were percolation theory, random graphs, statisti- initially proposed by Heathcote [2,3] as cal mechanics, the analysis of algorithms, branching models that could have a non- and bins and balls problems, to name a few. The appearance of branching processes in trivial stationary distribution. Lyons et al. so many contexts has triggered the need for [4] showed, via change of measure argu- extensions and variations around the classi- ments, that they played a crucial role in the cal Galton—Watson branching process. For study of Galton—Watson processes. Finally, instance, their application in particle physics population-size-dependent branching pro- provided an impetus to study them in con- cesses, originally proposed by Labkovskii tinuous time. The possible extensions are [5], find their motivations in population almost endless, and indeed new models of dynamics: they are elegant models that processes exhibiting a branching structure introduce dependency between individuals are frequently proposed and studied. Such and can account for the important biological models allow for instance time and/or space notion of carrying capacity, see for instance to be continuous, individuals to have one [6–8]. The interested reader can find more of several types, immigration to take place, results in the extensive survey by Vatutin catastrophes to happen, individuals to move and Zubkov [9,10] that gathers results up to in space, each individual’s dynamic to depend 1993 as well as in the recent books by Haccou on time, space, the state of the process itself et al. [11] and by Kimmel and Axelrod [12]. or some exogenous resources, a combination Before going on, recall (see Intro- of all these ingredients, and many more. duction to Branching Processes)that In this article, we focus more specifically a Galton—Watson branching process ≥ N on three advanced models of branching (Zn, n 0) is an -valued Markov chain obeying to the following recursion: Wiley Encyclopedia of Operations Research and Management Science, edited by James J. Cochran Copyright © 2013 John Wiley & Sons, Inc. 1 2 ADVANCED BRANCHING PROCESSES extinction probability is a random variable in Zn the quenched approach, and a deterministic Zn+1 = Xni, n = 0, 1, 2, ...,(1)number in the annealed approach. k=1 When the environmental process is assumed to be stationary and ergodic, which = where the (Xni, n, i 0, 1, 2, ...)areinde- includes for instance the case of i.i.d. environ- pendent and identically distributed (i.i.d.) ment or the case where the environment is a random variables following the so-called stationary Markov chain, it is known since offspring distribution. A Galton—Watson the pioneering works of Smith [13], Smith process is classified according to the value and Wilkinson [14], and Athreya and Karlin = E of the mean m (Xni) of its offspring [15,16] that the extinction problem and the distribution. If m < 1, the process is subcrit- description of the asymptotic growth have ical: it dies out almost surely, the survival fairly general solutions. Although in the clas- probability P(Zn > 0) decays exponentially n sical Galton—Watson case, the classification fast at speed m ,andZn conditioned on being of Z is in terms of the mean of the offspring = n non-zero converges weakly. If m 1, the pro- distribution, it is not difficult to see that in cess is critical: it dies out almost surely, the the case of random (stationary and ergodic) P > survival probability (Zn 0) decays poly- environment the mean of the logarithm of nomially fast, and Zn conditioned on being the mean is the meaningful quantity to non-zero grows polynomially fast. Finally, look at. More precisely, if π is a probability if m > 1, the process is supercritical: it may distribution on N,letm(π) = yπ({y})be survive forever, and grows exponentially fast y its mean. Then, by definition (1), we have in the event {∀n ≥ 0:Zn > 0} of survival. Sn E (Zn | ) = Z0m(π1) ···m(πn) = Z0e , BRANCHING PROCESSES IN RANDOM ENVIRONMENT wherewehavedefined A first possible generalization of the Sn = log m(π1) +···+log m(πn). Galton—Watson model allows for the off- spring distribution to vary over time: then, By the ergodic theorem, we have Sn/n → E →+∞ the recursion (1) still holds, the Xni s are still (log m (π1)) as n , which implies 1/n independent but the law of Xni may depend that E(Zn | ) → exp[E(log m(π1))]. In on n.Ifπn+1 is the offspring distribution in particular, conditionally on the environment, generation n and = (πn), that is, πn+1 is the mean of Zn goes to 0 if E(log m(π1)) < 0 the common law of the (Xni, i = 0, 1, 2, ...) and to +∞ if E(log m(π1)) > 0. This suggests and is the environmental process, then to classify the behavior of Zn in terms of this model defines a branching process in E(log m(π1)), and under some mild technical varying environment . We talk about assumptions it holds indeed that Zn dies out branching process in random environment almost surely if E(log m(π1)) ≤ 0 (subcritical when the sequence is itself random and and critical cases) and has a positive chance independent from Z0. Note that in this case, of surviving if E(log m(π1)) > 0 (supercritical πn is a random probability distribution on N. case). More precisely, we have the following As always in the case of stochastic quenched result: if q() is the (random) processes in random environment, one may extinction probability of Zn given ,then follow two approaches for their study: (i) the P(q() = 1) = 1 in the former case and quenched approach, which fixes a realization P(q() < 1) = 1inthelattercase. of the environment and studies the process In the supercritical case E(log m(π1)) > 0, in it; it is most natural from the point of there is an interesting technical condition view of the applications and (ii) the annealed that is both necessary and sufficient to approach, where the various characteristics allow the process to survive with posi- of interest are calculated by averaging tive probability: namely, in addition to over the environment. For instance, the E(log m(π1)) > 0 one also needs to assume ADVANCED BRANCHING PROCESSES 3 E(− log(1 − π1({0})) < +∞. This condition the behavior of its associated random walk. In shows the interesting interplay that arises particular, this work emphasized the major between Zn and the environment: even role played by fluctuation theory of random though E(log m(π1)) > 0 is sufficient to make walks in the study of branching processes in the conditional mean of Zn diverge, if random (i.i.d.) environment, a line of thought E(− log(1 − π1({0})) =+∞ then the process that has been very active since then. almost surely dies out because the probabil- Let us illustrate this idea with some of ity of having an unfavorable environment the results of Afanasyev et al. [17], so con- is large, where by unfavorable environment sider Zn a critical branching process in ran- we mean an environment π where the dom environment. As Zn is absorbed at 0, (random) probability π({0})ofhavingno we have P(Zn > 0 | ) ≤ P(Zm > 0 | )forany offspring is close to 1. In other words, if m ≤ n and as Zn is integer-valued, we obtain P | ≤ E | E(− log(1 − π1({0})) =+∞ then the process (Zn > 0 ) (Zm ). It follows that gets almost surely extinct because of the wide fluctuation of the environment. P(Zn > 0 | ) ≤ Z0 exp min Sm , 0≤m≤n The classification of Zn into the subcrit- ical, critical, and supercritical cases also which gives an upper bound, in term of the corresponds to different asymptotic behav- infimum process of the random walk S ,on iors of Z conditioned on non-extinction n n the decay rate of the extinction probability (here again, we have the quenched results in the quenched approach. It turns out that of Athreya and Karlin [15] in mind). In that this upper bound is essentially correct, and respect, Zn shares many similarities with that the infimum also leads to the correct a Galton—Watson process, although there decay rate of the extinction probability in the are some subtle differences as we see at the annealed approach, although in a different end of this section.
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