Catalytic branching processes via spine techniques and renewal theory
CCATALYTIC BRANCHING PROCESSES VIA SPINE TECHNIQUES ANDRENEWAL THEORY
LEIF D ¨ORING AND MATTHEW ROBERTS
Abstract.
In this article we contribute to the moment analysis of branching processes in cat-alytic media. The many-to-few lemma based on the spine technique is used to derive a systemof (discrete space) partial differential equations for the number of particles in a variation ofconstants formulation. The long-time behavior is then deduced from renewal theorems andinduction. Introduction and Results
A classical subject of probability theory is the analysis of branching processes in discrete orcontinuous time, going back to the study of extinction of family names by Francis Galton. Therehave been many contributions to the area since, and we present here an application of a recentdevelopment in the probabilistic theory. We identify qualitatively different regimes for the longtimebehaviour for moments of sizes of populations in a simple model of a branching Markov processin a catalytic environment.To give some background for the branching mechanism, we recall the discrete-time Galton-Watsonprocess. Given a random variable X with law µ taking values in N , the branching mechanism ismodelled as follows: for a deterministic or random initial number Z ∈ N of particles, one definesfor n = 1 , , ... Z n +1 = Z n (cid:88) r =0 X r ( n ) , where all X r ( n ) are independent and distributed according to µ . Each particle in generation n is thought of as giving birth to a random number of particles according to µ , and these parti-cles together form generation n + 1. For the continuous-time analogue each particle carries anindependent exponential clock of rate 1 and performs its breeding event once its clock rings.It is well-known that a crucial quantity appearing in the analysis is m = E [ X ], the expectednumber of offspring particles. The process has positive chance of long-term survival if and only if m >
1. This is known as the supercritical case. The cases m = 1 (critical) and m < t →∞ e − k ( m − t E (cid:2) Z kt (cid:3) ∈ (0 , ∞ ) ∀ k ∈ N if m < , (1.1) lim t →∞ t k − E (cid:2) Z kt (cid:3) ∈ (0 , ∞ ) ∀ k ∈ N if m = 1(1.2) lim t →∞ e − k ( m − t E (cid:2) Z kt (cid:3) ∈ (0 , ∞ ) ∀ k ∈ N if m > m >
1, increase polynomially if m = 1,and decay exponentially fast to zero if m < Mathematics Subject Classification.
Primary 60J27; Secondary 60J80.
Key words and phrases.
Renewal Theorem, Local Times, Branching Process, Many-to-Few Lemma.LD acknowledges the support of the Fondation Sciences Math´ematiques de Paris. MR thanks ANR MADCOF(grant ANR-08-BLAN-0220-01) and WIAS for their support. a r X i v : . [ m a t h . P R ] M a r LEIF D ¨ORING AND MATTHEW ROBERTS
In the present article we are interested in a simple spatial version of the Galton-Watson processfor which a system of branching particles moves in space and particles branch only in the presenceof a catalyst. More precisely, we start a particle ξ which moves on some countable set S accordingto a continuous-time Markov process with Q-matrix A . This particle carries an exponential clockof rate 1 that only ticks if ξ is at the same site as the catalyst, which we assume sits at some fixedsite 0 ∈ S . If and when the clock rings, then the particle dies and is replaced in its position by arandom number of offspring. This number is distributed according to some offspring distribution µ , and all newly born particles behave as independent copies of their parent: they move on S according to A and branch after an exponential rate 1 amount of time spent at 0.In recent years several authors have studied such branching systems. Often the first quantitiesthat are analyzed are moments of the form M k ( t, x, y ) = E (cid:2) N t ( y ) k (cid:12)(cid:12) ξ = x (cid:3) and M k ( t, x ) = E (cid:2) N kt (cid:12)(cid:12) ξ = x (cid:3) , where N t ( y ) is the number of particles alive at site y at time t , and N t = (cid:80) y ∈ S N t ( y ) is the totalnumber of particles alive at time t . Under the additional assumption that A = ∆ is the discreteLaplacian on Z d , the moment analysis was first carried out in [ABY98], [ABY98b], [AB00] viapartial differential equations and Tauberian theorems. More recently, the moment analysis, andmoreover the study of conditional limit theorems, was pushed forward to more general spatialmovement A assuming (A1) irreducibility, (A2) spatial homogeneity, (A3 ) symmetry, (A4 ) finite variance of jump sizes.Techniques such as Bellman-Harris branching processes (see [TV03],[VT05], [B10], [B11]), operatortheory (see [Y10]) and renewal theory (see [HVT10]) have been applied successfully. Some of thesetools also apply in a non-symmetric framework. We present a purely stochastic approach avoidingthe assumptions (A1)-(A4) . In order to avoid many pathological special cases we only assume (A) the motion governed by A is irreducible . This assumption is not necessary, and the interested reader may easily reconstruct the additionalcases from our proofs.In order to analyze the moments M k one can proceed in two steps. First, a set of partial differentialequations for M k is derived. This can be done for instance as in [AB00] via analytic arguments frompartial differential equations for the generating functions E x [ e − zN t ( y ) ] and E x [ e − zN t ] combinedwith Fa`a di Bruno’s formula of differentiation. The asymptotic properties of solutions to thosedifferential equations are then analyzed in a second step where more information on the transitionprobabilities corresponding to A implies more precise results on the asymptotics for M k . This iswhere the finite variance assumption is used via the local central limit theorem.The approach presented in this article is based on the combinatorial spine representation of [HR11]to derive sets of partial differential equations, in variation of constants form, for the k th momentsof N t ( y ) and N t . A set of combinatorial factors can be given a direct probabilistic explanation,whereas the same factors appear otherwise from Fa`a di Bruno’s formula. Those equations arethen analyzed via renewal theorems. We have to emphasize that under the assumption (A) only,general precise asymptotic results are of course not possible so that we aim at giving a qualitativedescription. Compared to the fine results in the presence of a local central limit theorem (such asLemma 3.1 of [HVT10] for finite variance transitions on Z ) our qualitative description is ratherpoor. On the other hand, the generality of our results allows for some interesting applications.For example, one can easily deduce asymptotics for moments of the number of particles whenthe catalyst is not fixed at zero, but rather follows some Markov process of its own, simply byconsidering the difference walk. ATALYTIC BRANCHING PROCESSES VIA SPINE TECHNIQUES AND RENEWAL THEORY 3
To state our main result, we denote the transition probabilities of A by p t ( x, y ) = P x ( ξ t = y ) andthe Green function by G ∞ ( x, y ) = (cid:90) ∞ p t ( x, y ) dt. Recall that, by irreducibility, the Green function is finite for all x, y ∈ S if and only if A istransient. For the statement of the result let us further denote by L t ( y ) = (cid:90) t { ξ s = y } ds the time of ξ spent at site y up to time t . Theorem 1.
Suppose that µ has finite moments of all orders; then the following regimes occurfor all integers k ≥ : i) If the branching mechanism is subcritical , then lim t →∞ M k ( t, x ) ∈ (0 , ∞ ) if A is transient , lim t →∞ M k ( t, x ) = 0 if A is recurrent , and lim t →∞ M k ( t, x, y ) = 0 in all cases. ii) If the branching mechanism is critical , then lim t →∞ M k ( t, x ) E x [ L t (0) k − ] ∈ (0 , ∞ ) and M ( t, x, y ) = p t ( x, y ) . iii) If the branching mechanism is supercritical , then there is a critical constant β = 1 G ∞ (0 ,
0) + 1 ≥ such that a) for m < β lim t →∞ M ( t, x ) ∈ (0 , ∞ ) and lim t →∞ M ( t, x, y ) = 0; further, there exist constants c and C such that c E x [ L t (0) k − ] ≤ M k ( t, x ) ≤ Ct k − . b) for m = β lim t →∞ M k ( t, x ) = ∞ , and lim t →∞ M k ( t, x, y ) = ∞ if (cid:90) ∞ rp r (0 , dr = ∞ lim t →∞ M k ( t, x, y ) t k − ∈ (0 , ∞ ) if (cid:90) ∞ rp r (0 , dr < ∞ . (In both cases the growth is subexonential.) c) for m > β lim t →∞ e − kr ( m ) t M k ( t, x, y ) ∈ (0 , ∞ ) and lim t →∞ e − kr ( m ) t M k ( t, x ) ∈ (0 , ∞ ) where r ( m ) equals the unique solution λ to (cid:82) ∞ e − λt p t (0 , dt = m − . LEIF D ¨ORING AND MATTHEW ROBERTS
We did not state all the asymptotics in cases ii) and iii)a). Our methods, see Lemma 3, do allowfor investigation of these cases too; in particular they show how M k ( t, x, y ) can be expressed recur-sively by M i ( t, x, y ) for i < k . However, without further knowledge of the underlying motion, it isnot possible to give any useful and general information. If more information on the tail of p t ( x, y )is available then the recursive equations can indeed be analyzed: for instance for kernels on Z d with second moments the local central limit theorem can be applied leading to p t ( x, y ) ∼ Ct − d/ ,and such cases have already been addressed by other authors.The formulation of the theorem does not include the limiting constants. Indeed, the proofs givesome of those (in an explicit form involving the transition probabilities p t ) in the supercriticalregime but they seem to be of little use. The use of spectral theory for symmetric Q -matrices A allows one to derive the exponential growth rate r ( m ) as the maximal eigenvalue of a Schr¨odingeroperator with one-point potential and the appearing constants via the eigenfunctions. Our renewaltheorem based proof gives the representation of r ( m ) as the inverse of the Laplace transform of p t (0 ,
0) at 1 / ( m −
1) and the eigenfunction expressed via integrals of p t (0 , p t (0 ,
0) is rarelyknown explicitly, the integral form of the constants is not very useful (apart from the trivial caseof Example 1 below). Only in case iii) b) for (cid:82) ∞ rp r (0 , dr = ∞ are the proofs unable to givestrong asymptotics. This is caused by the use of an infinite-mean renewal theorem which onlygives asymptotic bounds up to an unknown factor between 1 and 2. There is basically one examplein which p t (0 ,
0) is trivially known:
Example 1:
For the trivial motion A = 0, i.e. branching particles are fixed at the same site asthe catalyst, the supercritical cases iii) a) and b) do not occur as A is trivially recurrent so that β = 1. Furthermore, in this example p t (0 ,
0) = 1 for all t ≥ r ( m ) = m −
1. In fact byexamining the proof of Theorem 1 one recovers (1.1,1.2,1.3) with all constants.The explicit representation for the exponential growth rate allows for a more careful comparisonwith the non-spatial case.
Corollary 1.
Let r ( m ) be the exponential growth rate obtained in the supercritical case of Theo-rem 1. Then m (cid:55)→ r ( m ) is convex, with r (1) = 0 , r ( m ) ≤ m − , lim m →∞ r ( m )( m −
1) = 1 . Proof.
This follows from elementary manipulations of the defining equation for r ( m ). (cid:3) Remark 1.
We reiterate here that our results can be generalized when the fixed branching sourceis replaced by a random branching source moving according to a random walk independent of thebranching particles. For the proofs the branching particles only have to be replaced by branchingparticles relative to the branching source. Proofs
The key tool in our proofs will be the many-to-few lemma proved in [HR11] which relies on modernspine techniques. These emerged from work of Kurtz, Lyons, Pemantle and Peres in the mid-1990s[KLPP97, L97, LPP95]. The idea is that to understand certain functionals of branching processes,it is enough to carefully study the behaviour of one special particle, the spine . In particular verygeneral many-to-one lemmas emerged, allowing one to easily calculate expectations of sums overparticles like E (cid:34) (cid:88) v ∈ N t f ( v ) (cid:35) , where f ( v ) is some well-behaved functional of the behaviour of the particle v up to time t , and N t here is viewed as the set of particles alive at time t , rather than the number. It will always beclear from the context which meaning for N t is intended. ATALYTIC BRANCHING PROCESSES VIA SPINE TECHNIQUES AND RENEWAL THEORY 5
It is natural to ask whether similar results exist for higher moments of sums over N t . This is theidea behind [HR11], wherein it turns out that to understand the k th moment one must considera system of k particles. The k particles introduce complications compared to the single particlerequired for first moments, but this is still significantly simpler than controlling the behaviour ofthe potentially huge random number of particles in N t .While we do not need to understand the full spine setup here, we shall require some explanation.For each k ≥ p k = P ( X = k ) and m k = E [ X k ], the k th moment of the offspring distribution(in particular m = m ). We define a new measure Q = Q kx , under which there are k distinguishedlines of descent known as spines. The construction of Q relies on a carefully chosen change ofmeasure, but we do not need to understand the full construction and instead refer to [HR11]. Inorder to use the technique, we simply have to understand the dynamics of the system under Q .Under Q kx particles behave as follows: • We begin with one particle at position x which (as well as its position) has a mark k . Wethink of a particle with mark j as carrying j spines. • Whenever a particle with mark j , j ≥
1, spends an (independent) exponential time withparameter m j in the same position as the catalyst, it dies and is replaced by a randomnumber of new particles with law A j . • The probability of the event { A j = a } is a j p a m − j . (This is the j th size-biased distributionrelative to µ .) • Given that a particles v , . . . , v a are born, the j spines each choose a particle to followindependently and uniformly at random. Thus particle v i has mark l with probability a − l (1 − a − ) j − l , l = 0 , . . . , j , i = 1 , . . . , a . We also note that this means that there arealways k spines amongst the particles alive; equivalently the sum of the marks over allparticles alive always equals k . • Particles with mark 0 are no longer of interest (in fact they behave just as under P ,branching at rate 1 when in the same position as the catalyst and giving birth to numbersof particles with law µ , but we will not need to use this).For a particle v , we let X v ( t ) be its position at time t and B v be its mark (the number of spines itis carrying). Let σ v be the time of its birth and τ v the time of its death, and define σ v ( t ) = σ v ∧ t and τ v ( t ) = τ v ∧ t . Let χ it be the current position of the i th spine. We call the collection ofparticles that have carried at least one spine up to time t the skeleton at time t , and write skel( t ).Figure 1 gives an impression of the skeleton at the start of the process. Figure 1.
An impression of the start of the process: each particle in the skeletonis a different colour, and particles not in the skeleton are drawn in pale grey. Thecircles show the number of spines being carried by each particle in the skeleton.A much more general form of the following lemma was proved in [HR11].
LEIF D ¨ORING AND MATTHEW ROBERTS
Lemma 1 (Many-to-few) . Suppose that f : R → R is measurable. Then, for any k ≥ , E (cid:88) v ,...,v k ∈ N t f ( X v ( t )) · · · f ( X v k ( t )) = Q k f ( χ t ) · · · f ( χ kt ) (cid:89) v ∈ skel( t ) exp (cid:32) ( m B v − (cid:90) τ v ( t ) σ v ( t ) ( X v ( s )) ds (cid:33) . Clearly if we take f ≡
1, then the left hand side is simply the k th moment of the number ofparticles alive at time t . The lemma is useful since the right-hand side depends on at most k particles at a time, rather than the arbitrarily large random number of particles on the left-handside.Having introduced the spine technique, we can now proceed with the proof of Theorem 1. We firstuse Lemma 1 for the case k = 1, which is simply the many-to-one lemma, to deduce two convenientrepresentations for the first moments: a Feynman-Kac expression and a variation of constantsformula. Indeed, the exponential expression equally works for other random potentials and, hence,is well known for instance in the parabolic Anderson model literature. More interestingly, thevariation of constants representation is most useful in the case of a one-point potential: it simplifiesto a renewal type equation. Understanding when those are proper renewal equations replaces thespectral theoretic arguments of [ABY98] and explains the different cases appearing in Theorem 1. Lemma 2.
The first moments can be expressed as M ( t, x ) = E x (cid:2) e ( m − (cid:82) t ( ξ r ) dr (cid:3) , (2.1) M ( t, x, y ) = E x (cid:2) e ( m − (cid:82) t ( ξ r ) dr y ( ξ t ) (cid:3) , (2.2) where ξ t is a single particle moving with Q-matrix A . Furthermore, these quantities fulfill M ( t, x ) = 1 + ( m − p t ( x, ∗ M ( t, , (2.3) M ( t, x, y ) = p t ( x, y ) + ( m − p t ( x, ∗ M ( t, , y ) , (2.4) where ∗ denotes ordinary convolution in t . For completeness we include a proof of these well-known relations. First let us briefly mentionwhy the renewal type equations occur naturally. The Feynman-Kac representation can be provedin various ways; we derive it simply from the many-to-few lemma. The Feynman-Kac formulathen leads naturally to solutions of discrete-space heat equations with one-point potential: (cid:40) ∂∂t u ( t, x ) = A u ( t, x ) + ( m − ( x ) u ( t, x ) u (0 , x ) = y ( x ) . Applying the variation of constants formula for solutions gives u ( t, x ) = P t u (0 , x ) + (cid:90) t P t − s ( m − ( x ) u ( s, x ) ds = p t ( x, y ) + ( m − (cid:90) t p t − s ( x, u ( s, x ) ds, where P t is the semigroup corresponding to A , i.e. P t f ( x ) = E x [ f ( ξ t )]. Proof of Lemma 2.
To prove (2.1) and (2.2) we apply the easiest case of Lemma 1: we choose k = 1 and f ≡ f ( z ) = y ( z ) for (2.2)). Since there is exactly one spine at all times,the skeleton reduces to a single line of descent. Hence m B v − m − t . Thus M ( t, x ) = Q x (cid:2) e ( m − (cid:82) t ( ξ r ) dr (cid:3) and M ( t, x, y ) = Q x (cid:2) e ( m − (cid:82) t ( ξ r ) dr y ( ξ t ) (cid:3) ATALYTIC BRANCHING PROCESSES VIA SPINE TECHNIQUES AND RENEWAL THEORY 7 which is what we claimed but with expectations taken under Q rather than the original measure P . However we note that the motion of the single spine is the same (it has Q-matrix A ) underboth P and Q , so we may simply replace Q with P , giving (2.1) and (2.2).The variation of constants formulas can now be derived from the Feynman-Kac formulas. We onlyprove the second identity, as the first can be proved similarly. We use the exponential series to get E x (cid:104) e ( m − (cid:82) t ( ξ r ) dr y ( ξ t ) (cid:105) = E x (cid:34) ∞ (cid:88) n =0 ( m − n n ! (cid:18)(cid:90) t ( ξ r ) dr (cid:19) n y ( ξ t ) (cid:35) = P x ( ξ t = y ) + E x (cid:34) ∞ (cid:88) n =1 ( m − n n ! (cid:90) t . . . (cid:90) t ( ξ r ) · · · ( ξ r n ) dr n . . . dr y ( ξ t ) (cid:35) = p t ( x, y ) + E x (cid:34) ∞ (cid:88) n =1 ( m − n (cid:90) t (cid:90) tr . . . (cid:90) tr n − ( ξ r ) · · · ( ξ r n ) dr n . . . dr dr y ( ξ t ) (cid:35) . The last step is justified by the fact that the function that is integrated is symmetric in allarguments and, thus, it suffices to integrate over a simplex. We can exchange sum and expectationand obtain that the last expression equals p t ( x, y ) + ( m − (cid:90) t ∞ (cid:88) n =1 ( m − n − (cid:90) tr . . . (cid:90) tr n − P x [ ξ r = 0 , . . . , ξ r n = 0] dr n . . . dr dr . Due to the Markov property, the last expression equals p t ( x, y ) + ( m − (cid:90) t p r ( x, ∞ (cid:88) n =1 ( m − n − (cid:90) tr . . . (cid:90) tr n − P [ ξ r − r = 0 , . . . , ξ r n − r = 0] dr n . . . dr dr and can be rewritten as p t ( x, y ) + ( m − (cid:90) t p r ( x, (cid:32) ∞ (cid:88) n =1 ( m − n − (cid:90) t − r . . . (cid:90) t − r r n − P [ ξ r = 0 , . . . , ξ r n = 0] dr n . . . dr (cid:33) dr . Using the same line of arguments backwards for the term in parentheses, the assertion follows. (cid:3)
Having derived variation of constants formulas, there are different ways to analyze the asymptoticsof the first moments. Assuming more regularity for the transition probablities, this can be doneas sketched in the next remark.
Remark 2.
Taking Laplace transforms L in t , one can transform (2.3), and similarly (2.4), intothe algebraic equation L M ( λ, x ) = 1 λ + ( m − L M ( λ, L p λ ( x, , λ > , which can be solved explicitly to obtain L M ( λ, x ) = 1 λ (1 − ( m − L p λ ( x, , λ > . (2.5) Assuming the asymptotics of p t ( x, are known for t tending to infinity (and are sufficiently reg-ular), the asymptotics of L p λ ( x, for λ tending to zero can be deduced from Tauberian theorems.Hence, from Equation (2.5) one can then deduce the asymptotics of L M ( λ, x ) as λ tends to zero.This, using Tauberian theorems in the reverse direction, allows one to deduce the asymptotics of M ( t, x ) for t tending to infinity.Unfortunately, to make this approach work, ultimate monotonicity and asymptotics of the type p t ( x, ∼ Ct − α are needed. This motivated the authors of [ABY98] to assume (A4) so that bythe local central limit theorem p t ( x, ∼ (cid:18) d π (cid:19) d/ t − d/ . LEIF D ¨ORING AND MATTHEW ROBERTS
As we did not assume any regularity for p t , the aforementioned approach fails in general. Weinstead use an approach based on renewal theorems recently seen in [DS10]. Proof of Theorem 1 for M . Taking into account irreducibility and the Markov property of A ,we see that the property “ (cid:82) ∞ ( ξ r ) dr = ∞ almost surely” does not depend on the startingvalue ξ . To prove case i), we simply apply dominated convergence to (2.1) and (2.2). If A istransient, then (cid:82) ∞ ( ξ r ) dr < ∞ almost surely and M ( t, x ) converges to a constant. On theother hand if A is recurrent, then (cid:82) ∞ ( ξ r ) dr = ∞ almost surely and M ( t, x ) →
0. In both cases M ( t, x, y ) →
0, because if A is transient then { ξ t = y } → A is recurrentthen M ( t, x, y ) ≤ M ( t, x ) → M ( t, x ) = 1 and M ( t, x, y ) = p t ( x, y ). Next, for regime iii) a) weexploit both the standard and the reverse H¨older inequality for p > M ( t, x, y ) ≥ E x (cid:2) e − (1 / ( p − m − (cid:82) t ( ξ r ) dr (cid:3) − ( p − p t ( x, y ) p , (2.6) M ( t, x, y ) ≤ E x (cid:2) e p ( m − (cid:82) t ( ξ r ) dr (cid:3) /p p t ( x, y ) ( p − /p . (2.7)In the recurrent case G ∞ (0 ,
0) = ∞ and thus β = 1, so this case has already been dealt within regime ii). Hence we may assume that A is transient so that (cid:82) ∞ ( ξ r ) dr < ∞ with positiveprobability. This shows that the expectation in the lower bound (2.6) converges to a finite constant.By assumption m − < β so that there is p > p ( m − < β . With this choice of p , part3) of Theorem 1 of [DS10] implies that also the expectation in the upper bound (2.7) convergesto a finite constant. In total this shows that Cp t ( x, y ) p ≤ M ( t, x, y ) ≤ C (cid:48) p t ( x, y ) ( p − /p and the claim for M ( t, x, y ) follows. For M ( t, x ) we can directly refer to Theorem 1 of [DS10].For regimes iii) b) and c) we give arguments based on renewal theorems. A closer look at thevariation of constants formula (2.4) shows that only for x = 0, M ( t, x, y ) occurs on both sidesof the equation. Hence, we start with the case x = 0 and afterwards deduce the asymptotics for x (cid:54) = 0.Let us begin with the simpler case iii) c). As mentioned above, in this case we may assume that A is transient so that (cid:82) ∞ p r (0 , dr < ∞ . Hence, dominated convergence ensures that the equation (cid:82) ∞ e − λt p t (0 , dt = 1 / ( m −
1) has a unique positive root λ , which we call r ( m ). The definitionof r ( m ) shows that U ( dt ) := ( m − e − r ( m ) t p t (0 , dt is a probability measure on [0 , ∞ ) andfurthermore e − r ( m ) t p t (0 , y ) is directly Riemann integrable. Hence the classical renewal theorem(see page 349 of [F71]) can be applied to the (complete) renewal equation f ( t ) = g ( t ) + f ∗ U ( t ) , with f ( t ) = e − r ( m ) t M ( t, , y ) and g ( t ) = e − r ( m ) t p t (0 , y ). The renewal theorem implies thatlim t →∞ f ( t ) = (cid:82) ∞ g ( s ) ds (cid:82) ∞ U (( s, ∞ )) ds ∈ (0 , ∞ )(2.8)so that the claim for M ( t, , y ) follows including the limiting constants.For iii) b), we need to be more careful as the criticality implies that ( m − (cid:82) ∞ p r (0 , dr = 1.Hence, the measure U as defined above is already a probability measure so that the variation ofconstants formula is indeed a proper renewal equation. The renewal measure U only has finitemean if additionally (cid:90) ∞ rp r (0 , dr < ∞ . (2.9)In the case of finite mean the claim follows as above from (2.8) without the exponential correction(i.e. r ( m ) = 0). Note that p t (0 , y ) is directly Riemann integrable as the case β > A is transient and p t (0 , y ) is decreasing. ATALYTIC BRANCHING PROCESSES VIA SPINE TECHNIQUES AND RENEWAL THEORY 9
If (2.9) fails, we need a renewal theorem for infinite mean variables. Iterating Equation (2.4)reveals the representation M ( t, , y ) = p t (0 , y ) ∗ (cid:88) n ≥ ( m − n p t (0 , ∗ n , (2.10)where ∗ n denotes n -fold convolution in t and p t (0 , y ) ∗ p t (0 , ∗ = p t (0 , y ). Note that convergenceof the series is justified by( m − n p t (0 , ∗ n ≤ (cid:18) ( m − (cid:90) t p r (0 , dr (cid:19) n and the assumption on m . Lemma 1 of [E73] now implies that (cid:88) n ≥ ( m − n p t (0 , ∗ n ≈ t ( m − (cid:82) t (cid:82) ∞ s p r (0 , drds (2.11)which tends to infinity as ( m − (cid:82) ∞ s p r (0 , dr → s → ∞ since we assumed that ( m − p r (0 ,
0) is a probability density in r . To derive from this observation the result for M ( t, , y ),note that the simple bound p t (0 , y ) ≤ M ( t, , y ) ≤ (cid:90) t (cid:88) n ≥ ( m − n p r (0 , ∗ n dr. (2.12)For a lower bound, we use that due to irreducibility and continuity of p t (0 , y ) in t , there are0 < t < t and (cid:15) > p t (0 , y ) > (cid:15) for t ≤ t ≤ t . This shows that M ( t, , y ) ≥ (cid:15) (cid:90) t − t t − t (cid:88) n ≥ ( m − n p r (0 , ∗ n dr. (2.13)Combined with (2.11) the lower and upper bounds directly prove the claim for M ( t, , y ).It remains to deal with regime iii) b) and c) for x (cid:54) = 0. The results follow from the asymptoticsof the convolutions as those do not vanish at infinity. But this can be deduced from simple upperand lower bounds similar to (2.12) and (2.13).The asymptotic results for the expected total number of particles M ( t, x ) follow from similarideas: estimating as before1 + (cid:15) (cid:90) t − t t − t M ( r, dr ≤ M ( t, x ) ≤ (cid:90) t M ( r, dr, and applying case 2) of Theorem 1 of [DS10] to (2.1) with x = 0, the result follows. (cid:3) We now come to the crucial lemma of our paper. We use the the many-to-few lemma to reducehigher moments of N t and N t ( y ) to the first moment. More precisely, a system of equations isderived that can be solved inductively once the first moment is known. This particular usefulform is caused by the one-point catalyst. A similar system can be derived in the same manner inthe deterministic case if the one-point potential is replaced by a n -point potential. However thecase of a random n -point potential is much more delicate as the sources are “attracted” to theparticles, destroying any chance of a renewal theory approach. Lemma 3.
For k ≥ the k th moments fulfill M k ( t, x ) = M ( t, x ) + M ( t, x, ∗ g k (cid:0) ( M ( t, , · · · , M k − ( t, (cid:1) , (2.14) M k ( t, x, y ) = M ( t, x, y ) + M ( t, x, ∗ g k (cid:0) M ( t, , y ) , · · · , M k − ( t, , y ) (cid:1) , (2.15) where g k (cid:0) M , ..., M k − (cid:1) = k (cid:88) j =2 E (cid:20)(cid:18) Xj (cid:19)(cid:21) (cid:88) i ,...,i j > i + ... + i j = k k ! i ! · · · i j ! M i · · · M i j . Proof.
We shall only prove equation (2.14); the proof of equation (2.15) is almost identical. Werecall the spine setup and introduce some more notation. To begin with, all k spines are carriedby the same particle ξ which branches at rate m k = E [ X k ] when at 0. Thus the k spines separateinto two or more particles at rate m k − m when at 0 (since it is possible that at a birth event all k spines continue to follow the same particle, which happens at rate m ). We consider what happensat this first “separation” time, and call it T .Let i , . . . , i j > i + . . . + i j = k , and define A k ( j ; i , . . . , i j ) to be the event that at a separationevent, i spines follow one particle, i follow another, . . . , and i j follow another. The first particlesplits into a new particles with probability a k p a m − k (see the definition of Q k ). Then given thatthe first particle splits into a new particles, the probability that i spines follow one particle, i follow another, . . . , and i j follow another is1 a k · (cid:18) aj (cid:19) · k ! i ! · · · i j !(the first factor is the probability of each spine making a particular choice from the a available;the second is the number of ways of choosing the j particles to assign the spines to; and the thirdis the number of ways of rearranging the spines amongst those j particles). Thus the probabilityof the event A k ( j ; i , . . . , i j ) under Q k is1 m k E (cid:20)(cid:18) Xj (cid:19)(cid:21) k ! i ! · · · i j ! . (Note that, as expected, this means that the total rate at which a separation event occurs is m k · m k k (cid:88) j =2 E (cid:20)(cid:18) Xj (cid:19)(cid:21) (cid:88) i ,...,i j > i + ... + i j = k k ! i ! · · · i j ! = m k − m since the double sum is just the expected number of ways of assigning k things to X boxes withoutassigning them all to the same box.)However, for j ≥ given that we have a separation event, A k ( j ; i , . . . , i j ) occurs with probability1 m k E (cid:20)(cid:18) Xj (cid:19)(cid:21) k ! i ! · · · i j ! (cid:18) m k m k − m (cid:19) . Write χ t for the position of the particle carrying the k spines for t ∈ [0 , T ), and define F t to bethe filtration containing all information (including about the spines) up to time t . Recall that theskeleton skel( t ) is the tree generated by particles containing at least one spine up to time t ; letskel( s ; t ) similarly be the part of the skeleton falling between times s and t . Using the many-to-fewlemma with f = 1, the fact that by definition before T all spines sit on the same particle andintegrating out T , we obtain E (cid:2) N kt (cid:3) = Q k (cid:89) v ∈ skel( t ) e ( m Bv − (cid:82) τv ( t ) σv ( t ) ( X v ( s )) ds = Q k e ( m k − (cid:82) T ( χ s ) ds { T ≤ t } Q k (cid:89) v ∈ skel( T ; t ) e ( m Bv − (cid:82) τv ( t ) σv ( t ) ( X v ( s )) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F T + Q k (cid:104) e ( m k − (cid:82) t ( χ s ) ds { T >t } (cid:105) = (cid:90) t Q k (cid:34) e ( m k − (cid:82) u ( χ s ) ds ( m k − m ) ( χ u ) e − ( m k − m ) (cid:82) u ( χ s ) ds · Q k (cid:20) (cid:89) v ∈ skel( u ; t ) e ( m Bv − (cid:82) τv ( t ) σv ( t ) ( X v ( s )) ds (cid:12)(cid:12)(cid:12)(cid:12) F u ; T = u (cid:21)(cid:35) du + Q k (cid:104) e ( m k − (cid:82) t ( χ s ) ds e − ( m k − m ) (cid:82) t ( χ s ) ds (cid:105) . ATALYTIC BRANCHING PROCESSES VIA SPINE TECHNIQUES AND RENEWAL THEORY 11
To prove equation (2.15), the same arguments are used with f = y in place of f = 1. Now wesplit the sample space according to the distribution of the numbers of spines in the skeleton attime T . Since, given their positions and marks at time T , the particles in the skeleton behaveindependently, we may split the product up into j independent factors. Thus E (cid:2) N kt (cid:3) = (cid:90) t k (cid:88) j =2 (cid:88) i ,...,i j > i + ... + i j = k E (cid:20)(cid:18) Xj (cid:19)(cid:21) k ! i ! · · · i j ! Q k (cid:34) e ( m − (cid:82) u ( χ s ) ds ( χ u ) · j (cid:89) l =1 Q i l (cid:20) (cid:89) v ∈ skel( t − u ) e ( m Bv − (cid:82) τv ( t − u ) σv ( t − u ) ( X v ( s )) ds (cid:21)(cid:35) du + Q k (cid:104) e ( m − (cid:82) t ( χ s ) ds (cid:105) = (cid:90) t k (cid:88) j =2 (cid:88) i ,...,i j > i + ... + i j = k E (cid:20)(cid:18) Xj (cid:19)(cid:21) k ! i ! · · · i j ! E x [ N u (0)] · j (cid:89) l =1 E (cid:2) N i l t − u (cid:3) du + E x [ N t ] , where we have used the many-to-few lemma backwards with f = (first expectation) and f = 1(two last expectations) to obtain the last line. This is exactly the desired equation (2.14). ForEquation (2.15) we again use f = y in place of f = 1 and copy the same lines of arguments. (cid:3) Remark 3.
The factors appearing in g k are derived combinatorially from splitting the spines. InLemma 3.1 of [AB00] they appeared from Fa`a di Bruno’s differentiation formula. We need the following elementary lemma before we can complete our proof.
Lemma 4.
For any non-negative integer-valued random variable Y , and any integers a ≥ b ≥ , E [ Y a ] E [ Y ] ≥ E [ Y b ] E [ Y a − b +1 ] . Proof.
Assume without loss of generality that b ≥ a/
2. Note that for any two positive integers j and k , j a k + jk a − j b k a − b +1 − j a − b +1 k b = jk ( j − k ) (cid:16) j a − + 2 j a − k + 3 j a − k + . . . + ( a − b ) j b − k a − b − + ( a − b − j b − k a − b + . . . + 2 jk a − + k a − (cid:17) ≥ . Thus E [ Y a ] E [ Y ] − E [ Y b ] E [ Y a − b +1 ]= (cid:88) j ≥ j a P ( Y = j ) (cid:88) k ≥ k P ( Y = k ) − (cid:88) j ≥ j b P ( Y = j ) (cid:88) k ≥ k a − b +1 P ( Y = k )= (cid:88) j ≥ (cid:88) k>j ( j a k + k a j − j b k a − b +1 − j a − b +1 k b ) P ( Y = j ) P ( Y = k ) ≥ (cid:3) We can now finish the proof of the main result.
Proof of Theorem 1 for M k . Case i) follows just as for M , applying dominated convergence tothe Q k -expectation in Lemma 1. Note that if T is the first split time of the k spines (as in Lemma 3)then e ( m k − (cid:82) T ( ξ s ) ds is stochastically dominated by e ( m k − τ where τ is an exponential randomvariable of parameter m k − m ; this allows us to construct the required dominating randomvariable. For case ii), using Lemmas 2 and 3 we find the lower bound M k ( t, x ) ≥ C (cid:90) t p s ( x, M k − ( t − s, ds ≥ C (cid:90) t p s ( x, M k − ( t − s, ds. (2.16)An upper bound can be obtained by additionally using Lemma 4 (to reduce g k to the leading term M M k − ) to obtain M k ( t, x ) ≤ C (cid:90) t p s ( x, M k − ( t − s, ds. (2.17)Using inductively the lower bound (2.16) and furthermore the iteration (cid:90) t P x ( X s = 0) (cid:90) t − s P ( X s = 0) . . . (cid:90) t − s − ... − s k − P ( X s k − = 0) ds k − . . . ds ds = (cid:90) t (cid:90) ts . . . (cid:90) ts k − P x ( X s = 0 , X s = 0 , . . . , X s k − = 0) ds k − . . . ds ds = 1( k − (cid:90) t (cid:90) t . . . (cid:90) t P x ( X s = 0 , X s = 0 , . . . , X s k − = 0) ds k − . . . ds ds = 1( k − E x (cid:34)(cid:18)(cid:90) t { X s =0 } ds (cid:19) k − (cid:35) = 1( k − E x (cid:2) L t (0) k − (cid:3) (2.18)we see that M k ( t, x ) goes to infinity if A is recurrent and to a constant if A is transient. Thisimplies that the additional summand 1 in (2.17) can be omitted asymptotically in both cases. Theclaim follows.The lower bound of case iii)a) follows by the same argument as for case ii), and the upper boundis a straightforward induction using Lemmas 3 and 4. The cases iii)b) and c) also follow fromLemma 3 and induction based on the asymptotics for M . (cid:3) Acknowledgements
LD would like to thank Martin Kolb for drawing his attention to [ABY98] and Andreas Kyprianoufor his invitation to the Bath-Paris workshop on branching processes, where he learnt of the many-to-few lemma from MR. The authors would also like to thank Piotr Milos for checking an earlierdraft, and a referee for pointing out several relevant articles.
References [ABY98] Albeverio, S.; Bogachev, L.; Yarovaya, E. “Asyptotics of branching symmetric random walk on the latticewith a single source” C. R. Acad. Sci. Paris, 326, Serie 1, pp. 975-980, 1998[ABY98b] Albeverio, S.; Bogachev, L.; Yarovaya, E. “Eratum to: Asyptotics of branching symmetric random walkon the lattice with a single source” C. R. Acad. Sci. Paris, 326, Serie 1, pp. 975-980, 1998[AB00] Albeverio, S.; Bogachev, L. “Branching Random Walk in a Catalytic Medium. I. Basic Equations” Positivity4: 41100, 2000[AA87] Anderson, K. K.; Athreya, K. B. “A Renewal Theorem in the Infinite Mean Case” Annals of Probability,15, (1987), 388-393[B11] Bulinskaya E.V. “Limit distributions arising in branching random walks on integer lattices” Lithuan. Math.J., 2011, 51(3), p. 310-321[B10] Bulinskaya E.V. “Catalytic branching random walk on three-dimensional lattice” Theory Stoch. Process.,16(2):23-32, 2010[DS10] D¨oring, L.; Savov, M. “An Application of Renewal Theorems to Exponential Moments of Local Times”Elect. Comm. in Probab. 15 (2010), 263-269[E73] Erickson, B. “The strong law of large numbers when the mean is undefined” Trans. Amer. Math. Soc., 54,(1973), 371-381[F71] Feller, W. “An introduction to probability theory and its applications. Vol. II” John Wiley & Sons, Inc., NewYork-London-Sydney, (1966)[GH06] G¨artner, J.; Heydenreich, M. “Annealed Asymptotics for the Parabolic Anderson model with a MovingCatalyst” Stoch. Processes and Appl., 116, (2006), pp. 1511-1529
ATALYTIC BRANCHING PROCESSES VIA SPINE TECHNIQUES AND RENEWAL THEORY 13 [HR11] Harris, S.; Roberts, M. “The many-to-few lemma and multiple spines” arXiv:1106.4761v1[HVT10] Hu, Y.; Vatutin, V.A.; Topchii, V.A. “Branching random walk in Z with branching at the origin only”arXiv:1006.4769v1[KLPP97] Kurtz, T.; Lyons, R.; Pemantle, R.; Peres, Y. “A conceptual proof of the Kesten-Stigum theorem formulti-type branching processes” In K. B. Athreya and P. Jagers, editors, Classical and modern branchingprocesses (Minneapolis, MN, 1994) , volume 84 of
IMA Vol. Math. Appl. , pages 181–185. Springer, New York,1997.[L97] Lyons, R. “A simple path to Biggins’ martingale convergence for branching random walk” In K. B. Athreyaand P. Jagers, editors,
Classical and modern branching processes (Minneapolis, MN, 1994) , volume 84 of
IMAVol. Math. Appl. , pages 217–221. Springer, New York, 1997.[LPP95] Lyons, R.; Pemantle, R.; Peres, Y. “Conceptual proofs of L log L criteria for mean behavior of branchingprocesses” Ann. Probab. , 23(3):1125–1138, 1995.[TV03] Topchii, V. A.; Vatutin, V.A. “Individuals at the origin in the critical catalytic branching random walk”
Discrete Math. Theor. Comput. Sci. , 6:325-332, 2003.[VT05] Vatutin, V.A.; Topchii, V.A. “Limit theorem for critical catalytic branching random walks” Theory Prob.Appl., 2005, Vol. 49, No. 3, pp. 498-518[VTY04] Vatutin V.A.; Topchii V.A.; Yarovaya E.B. “Catalytic branching random walk and queueing systems withrandom number of independent servers” Theory Probab. Math. Stat., 69:1-15, 2004.[Y91] Yarovaya, E.B. “Use of spectral methods to study branching processes with diffusion in a noncompact phasespace” Teor. Mat. Fiz. 88 (1991), 25-30 (in Russian); English translation: Theor. Math. Phys. 88 (1991)[Y10] Yarovaya, E. B. “The monotonicity of the probability of return into the source in models of branching randomwalks” Mosc. Univ. Math. Bull., 65(2):78-80, 2010.
Fondation Math´ematique de Parisand Laboratoire de Probabilit´es et Mod´eles Al´eatoires (CNRS UMR. 7599) Universit´e Paris 6 Pierreet Marie Curie, U.F.R. Math´ematiques, 4 place Jussieu, 75252 Paris Cedex 05, France
E-mail address : [email protected] Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany
E-mail address ::