"Advanced Branching Processes" In: Wiley Encyclopedia of Operations Research and Management Science

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ADVANCED BRANCHING processes: (i) branching processes in random PROCESSES environment, which are examples of branching 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 processes 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:

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 offspring 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 subcritical, 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. In the supercritical case, form. Indeed, it can be shown under fairly Zn grows exponentially fast in the event of general assumptions that non-extinction, whereas in the subcritical case, Z conditioned on being non-zero con- n P(Zn > 0) ∼ θP(min(S1, ..., Sn) > 0) (2) verges weakly to a non-degenerate random variable. In the critical case, Zn conditioned for some θ ∈ (0, ∞). Moreover, conditionally +∞ Sn on being non-zero converges weakly to ,a on {Zn > 0}, Zn/e converges weakly to a result that can be refined in the case of i.i.d. random variable W, almost surely finite and environment. strictly positive, showing that Sn essentially Indeed, the case where the (πi) are i.i.d. governs the growth rate of Zn. Finally, has been extensively studied. In this case, although it is natural to consider the growth Sn is a random walk and recent works have rate and extinction probability of the process highlighted the intimate relation between Zn Zn, one can also reverse the viewpoint and and Sn. In particular, the classification of Zn study the kind of environment that makes can be generalized as follows. It is known the process survive for a long time. And from random walk theory that, when one actually, the conditioning {Zn > 0} has a excludes the trivial case where Sn = S0 for strong impact on the environment: although every n, there are only three possibilities Sn oscillates, conditionally on {Zn > 0} the concerning the almost sure behavior of (Sn): process (Sk,0≤ k ≤ n) suitably rescaled can either it drifts to −∞, or it oscillates with be shown to converge to the meander of a =−∞ =+∞ lim inf n Sn and lim supn Sn ,or Levy´ process, informally, a Levy´ process con- it drifts to +∞. Then, without assuming that ditioned on staying positive. This provides the mean E(log m(π1)) exists, Zn can be said to another illustration of the richness of this be subcritical if Sn →−∞, critical if Sn oscil- class of models, where the interplay between lates, and supercritical if Sn →+∞.Within the environment and the process leads to this terminology, Afanasyev et al. [17] stud- very interesting behavior. ied the critical case and were able to obtain These various results concern the striking results linking the behavior of Zn to annealed approach: Equation (2) is for 4 ADVANCED BRANCHING PROCESSES instance obtained by averaging over the and attaining a high level, have been studied environment. However, the connection in Refs 24,25 and 26,27, respectively. In between Zn and Sn continues to hold in Ref. 28, the survival probability of the critical the quenched approach. In Ref. 18, it is for multitype branching process in a Markovian instance shown that Zn passes through a random environment is investigated. number of bottlenecks at the moments close to the sequential points of minima in the associated random walk. More precisely, if BRANCHING RANDOM WALKS τ(n) = min{k ≥ 0:Sj ≥ Sk, j = 0, ..., n} is the leftmost point of the interval [0, n]at Branching random walks are extension of which the minimal value of (Sj, j = 0, ..., n) Galton—Watson processes that, in addition is attained, Zτ(n) conditionally on the envi- to the genealogical structure, add a spatial ronment and on {Zn > 0} converges weakly to component to the model. Each individual has a finite random variable. For further reading a location, say for simplicity on the real line. on this topic, the reader is referred to Refs 19 Typically, each individual begets a random and 20. number of offspring, as in a regular branching Let us conclude this section by complet- process, and the positions of these children ing the classification of branching processes form a point process centered around the in random environment. We have mentioned location of the parent. For instance, if B = that similarly as Galton—Watson processes, x∈B δx is the law of the point process gov- branching processes in random environment erning the locations of the offspring of a given could be classified as subcritical, critical, or individual, with δx the Dirac measure at x ∈ supercritical according to whether E(Y) < 0, R, the locations of the children of an individ- E(Y) = 0, or E(Y) > 0, with Y = log m(π1) (in ual located at y are given by the atoms of the the ‘‘simple’’ case where Y is indeed inte- measure x∈B δy+x. Branching random walks grable). Interestingly, assuming that E(etY )is can therefore naturally be seen as measure- finite for every t ≥ 0, the subcritical phase can valued Markov processes, which will turn out be further subdivided, according to whether to be the right point of view when discussing E(YeY ) > 0, E(YeY ) = 0, or E(YeY ) < 0 corre- superprocesses. Another viewpoint is to see sponding respectively, in the terminology of branching random walks as random labeled Birkner et al. [21], to the weakly subcriti- trees: the tree itself represents the genealog- cal, intermediate subcritical, and strongly ical structure, whereas the label on an edge subcritical cases. These three cases corre- represents the displacement of a child with spond to different speeds of extinction: in respect to its parent. Nodes of the tree then the weakly subcritical case, there exists β ∈ naturally inherit labels recursively, where βY (0,1)suchthatE(Ye ) = 0andP(Zn > 0) the root is assigned any label, and the label decays like n−3/2[E(eβY )]n;intheinterme- of a node that is not the root is given by diate subcritical case, P(Zn > 0) decays like the label of its parent plus the label on the n−1/2[E(eY )]n; finally, in the strongly subcrit- corresponding edge. Y n ical case, P(Zn > 0) decays like [E(e )] .These There is an interesting connection decay rates are to be compared to the clas- between branching random walks and (gen- sical Galton—Watson process, wherein the eral) branching processes. In the case where n subcritical case P(Zn > 0) decays like m ,cor- the atoms of B are in (0, ∞), particles of the responding to the strongly subcritical case, branching random walk live on the positive because when Y is determinist we have the half-line and their positions can therefore relation m = EeY . be interpreted as the time at which the Further reading on branching processes in corresponding particle is born. Keeping track random environment includes, for example, of the filiation between particles, we see Refs 22 and 23 for the study of the subcritical that within this interpretation, each particle case using the annealed approach, while the gives birth at times given by the atoms of trajectories of Zn under various condition- the random measure B. This is exactly the ings, namely dying at a distant given moment model of general, or Crump—Mode—Jagers, ADVANCED BRANCHING PROCESSES 5 branching processes (see Introduction to with common distribution Sn equal to the Branching Processes). value at time n of a random walk with step One of the most studied questions related distribution D, we obtain for any θ ≥ 0 to branching random walks concerns the n long-term behavior of extremal particles. Of P (Mn ≤ an) ≤ 2 P (Sn ≤ an) course, as the branching random walk is − n ≤ 2n eθanE(e θSn ) ≤ 2μ(a) absorbed when there are no more particles, this question only makes sense when the using Markov inequality for the sec- underlying Galton—Watson process is ond inequality, and defining μ(a)as supercritical and conditioned on surviving. θa −θD μ(a) = infθ≥0 e E(e ) in the last term. In Let for instance Mn be the location of the particular, P(Mn/n ≤ a) → 0ifa is such that leftmost particle in the nth generation, that μ(a) < 1/2, which makes Mn/n → γ ,with is, the smallest label among the labels of γ = inf {a : μ(a) > 1/2}, the best we could all nodes at depth n in the tree. Assume hope for. It is quite surprising that these sim- for simplicity that each individual has two ple computations lead to the right answer, children with i.i.d. displacements, say with but the almost sure convergence Mn/n → γ distribution D. Then by construction, a is indeed the content of the aforementioned typical line of descent (i.e., the labels on the Hammersley—Kingman—Biggins theorem. successive nodes on an infinite path from the The case γ = 0 can be seen as a critical root) is equal in distribution to a random case, where the speed is sublinear; for a large walk with step distribution D, thus drifting class of branching random walks, this case +∞ E to if D > 0. However, Mn is then equal also corresponds, after some renormalization in distribution to the minimum between (typically, centering the branching random n 2 random walks, and although a typical walk), to study the second-order asymptotic line of descent goes to +∞, the exponential behavior of Mn for a general γ .Inthecase explosion in the number of particles makes it γ = 0, several asymptotic behaviors are possible for the minimal displacement Mn to possible and the reader can for instance follow an atypical trajectory and, say, diverge consult Addario-Berry and Reed [32] for more to −∞. Finer results are even available, details. Bramson [33] proved that if every and the speed at which Mn diverges has particle gives rise to exactly two particles and been initiated in a classical work by Ham- the displacement takes the value 0 or 1 with mersley [29] (who was interested in general equal probability, then Mn − log log n/ log 2 branching processes), and later extended by converges almost surely. Recently, A¨ıdekon´ Kingman [30] and Biggins [31] leading to [34] proved in the so-called boundary case what is now commonly referred to as the that Mn − (3/2) log n converges weakly. Hammersley—Kingman—Biggins theorem. These results concern the behavior of the For instance, in the simple case with binary extremal particle, and there has recently offspring and i.i.d. displacements, it can be been an intense activity to describe the −θD shown that Mn →−∞if infθ≥0E(e ) > 1/2 asymptotic behavior of all extremal particles, and simple computations even give a precise that is, the largest one, together with the idea of the speed at which this happens. second largest one, and third largest one. (k) = n n Indeed, if Sn for k 1, ...,2 are the 2 Informally, one is interested in the behavior labels of the nodes at depth n in the tree, we of the branching random walk ‘‘seen from its have by definition for any a ∈ R tip,’’ which technically amounts to consider the point process recording the distances P ≤ = P (k) ≤ ≤ n (Mn an) Sn an for some k 2 from every particle to the extremal one. This question was recently solved by Madaule ≤ P (k) ≤ Sn an [35], building on previous results by different 1≤k≤2n authors, in particular the aforementioned work by A¨ıdekon´ [34]. The limiting point using the union bound for the last inequality. process can be seen as a ‘‘colored’’ Poisson pro- (k) As the Sn ’s are identically distributed, say cess, informally obtained by attaching to each 6 ADVANCED BRANCHING PROCESSES atom of some Poisson process independent investigated in Refs 39–45. There are also a copies of some other point process. number of articles for the so-called catalytic Initially, one of the main motivations for random walk when the particles performing studying the extremal particle of a branching random walk on Zd reproduce at the origin random walk comes from a connection with only [42,46,47] for which a wide range of phe- the theory of partial differential equations. nomena has been investigated. For this and Namely, one can consider a variation of the more, the reader can for instance consult the branching random walk model, called the recent survey by Bertacchi and Zuccha [48]. branching Brownian motion. In this model, time and space are continuous; each particle lives for a random duration, exponentially CONTINUOUS STATE BRANCHING distributed, during which it performs a PROCESSES Brownian motion, and is replaced on death From a modeling standpoint, it is natural by k particles with probability pk. Then, McK- in the context of large populations to won- ean [36] and later Bramson [37] observed = 2 +∞ der about branching processes in continuous that if k kpk 2and k k pk < , then the function u(t, x) = P(M(t) > x), with time and with a continuous state space. In M(t) now the maximal displacement of the same vein, Brownian motion (or, more the branching Brownian motion at time generally, a Levy´ process) approximates a t, that is, the location of the rightmost random walk evolving on a long time scale. particle, is a solution to the so-called The definition (1) does not easily lend itself Kolmogorov—Petrovskii—Piskunov (KPP) to such a generalization. equation, which reads An alternative and, from this per- spective, more suitable characterization ∂u 1 ∂2u of Galton—Watson processes is through = + p uk − u 2 k the branching property. If Zy denotes ∂t 2 ∂x ≥ k 1 a Galton—Watson process started with ∈ N with the initial condition u(0, x) = 1ifx ≥ 0 y individuals, the family of processes y ∈ N and u(0, x) = 0ifx < 0. One of the key prop- (Z , y ) is such that erties of the KPP equation is that it admits + (d) traveling waves: there exists a unique solu- Zy z = Zy + Zz, y, z ∈ N,(3) tion satisfying where (=d) means equality in distribution and + → u t, m(t) x w(x)uniformly in x as Zz is a copy of Zz, independent from Zy. t →+∞. In words, a Galton—Watson process with offspring distribution X started with y + z Using the connection with the branching individuals is stochastically equivalent to the Brownian motion, Bramson [37] was able sum of two independent Galton—Watson to derive extremely precise results on the processes, both with offspring distribution position of the traveling√ wave, and essen- X, and where one starts with y individu- tially proved that m(t) = 2t − (3/23/2)logt. als and the other with z. It can actually be In probabilistic√ terms, this means that shown that this property uniquely charac- M(t) − 2t + (3/23/2)logt converges weakly. terizes Galton—Watson processes, and Lam- Similarly as for the branching random walk, perti [49] uses this characterization as the there has recently been an intense activity definition of a continuous state branching to describe the branching Brownian motion process (CSBP). Formally, CSBPs are the seen from its tip, which culminated in only time-homogeneous Markov processes (in the recent works by Arguin et al. [38] and particular, in continuous time) with state A¨ıdekon´ [34]. space [0, ∞] that satisfy the branching prop- Beyond the behavior of extremal parti- erty, see also Ikeda et al. [50] for more gen- cles, the dependency of the branching ran- eral state spaces. Note that even in the dom walk on the space dimension has been case of real-valued branching processes, the ADVANCED BRANCHING PROCESSES 7 state space includes +∞:incontrastwith that (λ) =−log E(e−λY(1)) or equivalently, Galton—Watson branching processes, CSBP in view of the Levy—Khintchine´ formula, can in principle explode in finite time. is of the form One of the achievements of the theory 1 is the complete classification of CSBPs, the (λ) = ε + αλ − βλ2 main result being a one-to-one correspon- 2 dence between CSBPs and Levy´ processes −λx + 1 − e − λx1{x<1} π(dx) with no negative jumps and, in full general- (0,∞) ity, possibly killed after an exponential time. ∧ This result has a long history dating back for some measure π satisfying (0,∞)(1 from Lamperti [49], for which the reader x2)π(dx) < +∞ and some parameters can find more details in the introduction of ε, β ≥ 0, and α ∈ R. Conversely, any of this Caballero et al. [51] (note that CSBPs were form is the branching mechanism of a CSBP, first considered by Jirinaˇ [52]). There are two and as Levy´ processes are characterized by classical ways to see this result. Until further their Levy´ exponent, this provides an alter- notice, let Z = (Z(t), t ≥ 0) be a CSBP started native view on the one-to-one correspondence at Z(0) = 1. between CSBPs and Levy´ processes. The first one is through random time- The long-term behavior of CSBPs was one change manipulations, and more specifically of the earliest questions studied by Grey [53], through the Lamperti transformation L that and they exhibit a richer range of behavior acts on positive functions as follows. If f : than Galton—Watson processes. First, Z is [0, ∞) → [0,∞), then L(f ) is defined implic- conservative, that is, P(Z(t) < +∞) = 1for L t = ≥ ≥ itly by (f )( 0 f ) f (t)fort 0, and explicitly every t 0, if and only if 1/ is integrable L = ◦ = { ≥ u } + by (f ) f κ with κ(t) inf u 0: 0 f > t . at 0 , a sufficient condition for this being Then, it can be proved that L(Z)isaLevy´ that (0) = 0and (0) > −∞.Further,we process with no negative jumps, stopped at 0 observe by differentiating the branching and possibly killed after an exponential time; equation (4) with respect to λ and using ´ E = ∂u E = − conversely, if Y is such a Levy process, then (Z(t)) ∂λ (t,0) that (Z(t)) exp( (0)t), L−1(Y) is (well defined and is) a CSBP. which suggests to classify a (conservative) The second way is more analytical. Let CSBP as subcritical, critical, or supercritical u(t, λ) =−log E(e−λZ(t)): then u satisfies the according to whether (0) > 0, (0) = 0, semigroup property u(s + t, λ) = u(s, u(t, λ)), or (0) < 0, respectively. This classification which leads, for small h > 0, to the approxi- turns out to be essentially correct for mation conservative processes, under the additional requirement β > 0: in this case, supercritical u(t + h, λ) − u(t, λ) = u(h, u(t, λ)) − u(0, u(t, λ)) processes may survive forever, with prob- −λ ∂u ability e 0 where λ0 is the largest root of ≈ h (0, u(t, λ)) =−h (u(t, λ)) the equation (λ) = 0, while critical and ∂t subcritical processes die out almost surely, =−∂u that is, the time inf {t ≥ 0:Z(t) = 0} is almost once one defines (λ) ∂t (0, λ). It can indeed be shown that u satisfies the so-called surely finite. = branching equation When β 0, the situation may be slightly different. indeed, for any CSBP, the extinc- ∂u tion probability P(∃t ≥ 0:Z(t) = 0) is strictly =− (u), (4) +∞ ∂t positive if and only if 1/ is integrable at and (λ) > 0forλ large enough; in this case, − with boundary condition u(0, λ) = λ. In par- the extinction probability is equal to e λ0 ticular, uniquely characterizes u, and thus with λ0 as discussed earlier. In particular, we Z, and it is called the branching mechanism may have a subcritical CSBP (with (0) > 0) ∞ of Z. Moreover, it can be proved that satisfying both β = 0and (1/ ) =+∞,in is a Levy´ exponent, that is, there exists a which case Z(t) → 0butZ(t) > 0 for every Levy´ process Y with no negative jumps such t ≥ 0. In other words, although Z vanishes, 8 ADVANCED BRANCHING PROCESSES in the absence of the stochastic fluctuations If a CSBP can be viewed as a continuous induced by β, it never hits 0. This behavior is approximation of a Galton—Watson process, to some extent quite natural, because the α it is natural to ask about the existence of term corresponds to a deterministic exponen- a corresponding genealogical structure. This tial decay (for (λ) = αλ we have Z(t) = e−αt) question was answered by Duquesne and Le and the jumps of Z are only positive, so one Gall [57], who for each CSBP Z exhibited a needs stochastic fluctuations in order to make process H, which they call height process, Z hit 0. such that Z is the local time process of H. We have mentioned in the beginning This question is intrinsically linked to the of this section the motivation for studying study of continuum random trees initiated by CSBPs as continuous approximations of Aldous [58,59]. As a side remark, note that Galton—Watson processes. This line of this genealogical construction plays a key role thought is actually present in one of the in the construction of the Brownian snake earliest articles by Lamperti [54] on the [60]. There has also been considerable inter- subject. In particular, CSBPs are the only est in CSBP allowing immigration of new possible scaling limits of Galton—Watson individuals: these processes were defined by processes, that is, if (Z(n), n ≥ 1) is a sequence Kawazu and Watanabe [61], their genealogy (n) = of Galton—Watson processes with Z0 n studied by Lambert [62] and the correspond- such that the sequence of rescaled processes ing continuum random trees by Duquesne (n) (n) (Z , n ≥ 1), where Z (t) = Z(n) /n for some [63]. ant normalizing sequence an, converges weakly Finally, let us conclude this section on to some limiting process Z,thenZ must be CSBPs by mentioning superprocesses. Super- a CSBP. And conversely, any CSBP can be processes are the continuous approximations realized in this way. of branching random walks, in the same vein There is, at least informally, an easy way as CSBPs are continuous approximations of to see this result, by extending the Lam- Galton—Watson processes. They were con- perti transformation at the discrete level of structed by Watanabe [64], and can techni- Galton—Watson processes. Indeed, consider cally be described as measure-valued Markov (S(k), k ≥ 0) a random walk with step dis- processes. Similarly as for the branching tribution X = X − 1 for some integer-valued Brownian motion, Dynkin [65] showed that random variable X, and define recursively superprocesses are deeply connected to par- Z0 = S(0) and Zn+1 = S(0) + S(Z1 +···+Zn) tial differential equations. The recent book ≥ = + + for n 0. Then, writing S(k) S(0) X1 by Li [66] offers a nice account on this topic. ···+ Xk with (Xk) i.i.d. copies of X ,wehave

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