Critical parameters of germ-monotone families of branching random walks

CRITICAL P ARAMETERS OF GERM-MONOTONE F AMILIES OF BRANCHING RANDOM W ALKS DANIELA BER T A CCHI AND F ABIO ZUCCA Abstract. W e introduce a broad class of families of branc hing random w alks on a set X . The processes in each family are parametrized b y a p ositiv e parameter λ and they are monotonically increasing in λ with respect to the germ order, a notion that extends classical stochastic domina- tion. W e define a general notion of critical parameter λ ( A ) asso ciated with a subset A ⊆ X and inv estigate how mo difications of repro duction la ws affect these critical parameters. Keyw ords : branc hing random walk, branching pro cess, germ order, critical parameters, lo cal sur- viv al, global surviv al, pure global surviv al phase. AMS sub ject classification : 60J05, 60J80. 1. Introduction A branc hing process, also kno wn as the Galton–W atson process (see [17]), is a stochastic process in whic h a particle dies and pro duces a random num b er of offspring according to a prescrib ed offspring distribution ρ , where ρ ( n ) denotes the probability that exactly n children are b orn. Differen t particles repro duce indep endently following the same law. The pro cess ma y either b ecome extinct (that is, no particles remain alive after some finite time) almost surely , or survive indefinitely with p ositive probabilit y . The extinction probabilit y can b e explicitly c haracterized in terms of the offspring distribution ρ . A branching random walk (BR W, hereafter) generalizes this mo del by asso ciating each particle with a location x ∈ X , where X is an at most coun table set. Although X is often interpreted as a spatial domain, it can also represen t a set of types (see, for example, [22]). Particles lo cated at a site x ∈ X are replaced b y a random num b er of offspring, which are then distributed among the sites in X . The repro duction law dep ends on the lo cation of the parent particle, and all particles repro duce indep enden tly . This class of processes has b een extensiv ely s tudied; see, for instance, [1, 14, 19] for early con- tributions. Bey ond the classical setting of Z d , branc hing random w alks hav e b een inv estigated on finite sets [28], on trees [2, 23, 27, 29, 31, 33] and in random environmen t [25, 26]. Branc hing random walks may b e defined either in discrete time, where each particle lives for exactly one generation and all offspring app ear in the next generation, or in contin uous time, where generations o verlap because particles ma y repro duce at differen t times during their lifetimes. In con- tin uous time, one often considers families of parametrized pro cesses in whic h the repro duction rates b et ween lo cations are fixed, and a parameter λ > 0 controls the ov erall repro ductiv e sp eed (larger v alues of λ corresp ond to shorter interv als b etw een successiv e repro duction even ts; see Section 2 for details). By considering just the n umber and p ositions of all offspring generated by each particle in their own lifetime, w e obtain a discrete-time BR W, known as discr ete-time c ounterp art of the con tinuous time BR W (see Section 3.1 for details). More precisely , the discrete-time counterpart is in fact a family of discrete-time BR Ws parametrized by λ > 0. This family is increasing monotoni- cally with resp ect to the sto c hastic domination, that is, if λ > λ ∗ then the λ -pro cess sto chastically dominates the λ ∗ -pro cess. In this pap er, we introduce a general notion of critical v alues for broad families of branching random w alks, encompassing discrete-time coun terparts of contin uous-time BR Ws as well as discrete- time counterparts of certain non-Mark ovian pro cesses (see Section 3.1). W e fo cus on families of 1 BR Ws that are parametrized by λ > 0 and are monotone with resp ect to the germ order (see Definitions 3.1 and 3.2). Such families are referred to as germ-monotone BR Ws (briefly GMBR Ws ). Since a BR W evolv es on an underlying spatial structure, its b ehavior is t ypically more complex than that of a classical branching pro cess. In particular, one may distinguish b et ween global behavior (on the en tire space X ) and lo cal behavior (at a giv en site or within a finite subset A ⊆ X ). Starting from a single particle at a site x ∈ X , exactly one of the follo wing scenarios o ccurs: the pro cess b ecomes extinct almost surely , it survives globally but not lo cally , it survives b oth globally and lo cally , but with differen t probabilities (non-strong lo cal surviv al), it survives b oth globally and lo cally with the same probabilit y (strong lo cal surviv al). W e emphasize that the absence of strong lo cal surviv al includes b oth non-strong lo cal surviv al and almos t sure lo cal extinction. A fixed GMBR W ma y display different behaviors dep ending on the v alue of the parameter λ . The threshold separating global extinction from global surviv al is known as the global critical parameter λ w , while the threshold separating lo cal extinction from lo cal surviv al is called the lo cal critical pa- rameter λ s . The subscripts w and s stand for “weak” and “strong,” which are often used as synonyms for global and lo cal surviv al, resp ectively . V arious authors ha ve addressed the identification of these critical parameters [31, 32] and established criteria for surviv al and extinction [3, 18, 24, 29, 30, 33]. Here we introduce more general critical parameters λ ( A ) dep ending on the set A where surviv al is considered (see equation (3.2)). The v alue λ ( A ) is a threshold for λ separating almost sure extinction in A and positive probabilit y of surviv al in A (see Definition 2.2). The classic critical parameter ma y b e recov ered: λ w = λ ( X ) and λ s = λ ( A ) where A is a finite, nonempty set. Once surviv al with p ositiv e probability is established, one ma y further inv estigate the computation of extinction probabilities; see [8, 4, 9, 20]. Another field of inv estigation is the study of the directions of diffusion of a surviving pro cess (see, for example, [10, 15, 16, 21]). In ecological or epidemiological applications, the ob jectiv e ma y b e either to promote surviv al (for instance, in the case of endangered sp ecies) or to induce extinction (as in disease or p est control). F or GMBR Ws, the key factor that gov erns the dynamics is the family of reproduction laws µ λ . T o eradicate an epidemic, one p ossible s trategy is to reduce λ : for example, wearing masks to limit the spread of respiratory infections suc h as COVID-19 serves this purp ose. When λ ≤ λ s , the infection ev entually disapp ears from every site, whereas if λ w < λ , it may still p ersist globally . An alternativ e strategy is to mo dify the reproduction la ws µ λ b y reducing them, with resp ect to germ order, on a subset A 0 . Suc h a mo dification may increase the critical parameters and consequen tly change the b eha vior of the pro cess for a fixed v alue of λ . As an example, assume that λ is fixed and that λ ( A ) is the critical parameters asso ciated with a given GMBR W and a subset A . After altering the repro duction laws, the corresp onding critical parameter may shift to λ ∗ ( A ) and. If, for example, λ ( A ) < λ < λ ∗ ( A ), then the mo dified pro cess becomes extinct at A regardless its global b ehavior, even though the original pro cess exhibited surviv al in A . The effect of lo c al mo dific ations (namely , mo difications confined in a finite set) on λ s and λ w for contin uous-tim BR Ws has been studied in [11]. In this pap er, we consider the general class of GMBR Ws and we in vestigate how the critical parameters λ ( A ) ma y v ary when the reproduction la ws are modified on a generic, not necessarily finite, subset A 0 (see Section 4), under the assumption that surviv al in A 0 implies surviv al in A with conditional probability one. A notable sp ecial case o ccurs when A 0 is finite (see Section 5); in this setting, w e are able to analyze λ ( A ) for every nonempt y subset A . The pap er is organized as follows. Section 2 presents the basic definitions and preliminary results that are used throughout the pap er. The main to ol introduced in this section is Theorem 2.3, which has b een progressively refined o ver the y ears (see [7, Theorem 3.3], [9, Theorem 2.4], and [4, The- orem 4.1]). Section 3 introduces the class of GMBR Ws and discusses several illustrative examples, including contin uous-time BR Ws and aging BR Ws (see Section 3.1). Section 4 is devoted to the main results of the pap er, which rely on Theorem 4.1. Earlier versions of these results app eared in [8, Theorem 4.2] and [9, Theorem 2.4], and are further generalized in the present work. Prop osi- tion 4.3 and Corollary 4.5 analyze the b eha vior of min( λ ( A ) , λ ( B )) b efore and after a mo dification. 2 In particular, they sho w that either λ ( A ) ≥ λ ( B ) holds, or λ ( A ) cannot increase as a consequence of the modification. The last result of the section, Prop osition 4.6, deals with max( λ ( A ) , λ ( B )). These results are extremely p o werful in the case of lo c al mo dific ations (mo difications in a finite set); Section 5 is devoted to this sp ecial and imp ortan t case. One of the main consequences is that either λ ( A ) = λ s or λ ( A ) cannot increase as a consequence of lo cal mo difications; this is a rather surpris- ing results since the existence of a strictly larger critical v alue, λ ( B ) > λ ( A ), implies that no lo cal mo dification can increase the critical v alue related to A . Section 5.1 illustrates how the results of [11] can b e recov ered and further strengthened. Finally , an example is presented in Section 6 which sho ws that the critical v alue λ ( A ) can indeed differ from the classic critical v alues λ w and λ s ; this example shows how λ s , λ ( A ) and λ w v ary under lo cal mo difications- 2. Basic definitions and preliminaries 2.1. Discrete-time Branching Random W alks and their surviv al/extinction. Given an at most countable set X , we define a discrete-time BR W as a pro cess { η n } n ∈ N , where η n ( x ) is the num b er of particles aliv e at x ∈ X at time n . The dynamics is describ ed as follows: let S X := { f : X → N : P y f ( y ) < ∞} and let µ = { µ x } x ∈ X b e a family of probability measures on the (countable) measurable space ( S X , 2 S X ). A particle of generation n at site x ∈ X liv es one unit of time; after that, a function f ∈ S X is chosen at random according to the la w µ x . This function describ es the num b er of children and their p ositions, that is, the original particle is replaced by f ( y ) particles at y , for all y ∈ X . The choice of f is indep endent for all breeding particles. The BR W is denoted by ( X , µ ). W e define the first moment matrix M = ( m xy ) x,y ∈ X of the pro cess, where m xy := P f ∈ S X f ( y ) µ x ( f ) represen ts the exp ected num b er of children sent to y by a particle living at x (briefly , the exp ected n umber of particles from x to y ). T o a generic discrete-time BR W w e associate a graph ( X , E µ ), where ( x, y ) ∈ E µ if and only if m xy > 0. In contrast, given a graph ( X , E ), we sa y that a BR W ( X , µ ) is adapted to the graph if m xy > 0 if and only if ( x, y ) ∈ E . W e say that there is a path of length n from x to y , if it is p ossible to find a finite sequence { x i } n i =0 (where n ∈ N ) such that x 0 = x , x n = y and ( x i , x i +1 ) ∈ E µ for all i = 0 , . . . , n − 1 (observ e that there is alw ays a path of length 0 from x to itself ). If there is a path from x to y then we write x → y ; whenev er x → y and y → x we write x ⇌ y . The equiv alence relation ⇌ induces a partition of X : the class [ x ] of x is called irr e ducible class of x . If the graph ( X , E µ ) is c onne cte d (that is, there is only one irreducible class), then w e sa y that the BR W is irr e ducible . Irreducibility implies that the progeny of any particle can spread to any site of the graph with p ositive probabilit y . W e consider initial configurations with only one particle placed at a fixed site x and we denote by P x the la w of the corresp onding pro cess. Evolution of pro cesses with more than one initial particle can b e obtained by sup erimposition. In the following, wpp is shorthand for “with p ositiv e probability” (although, when talking ab out surviv al, “wpp” will usually b e tacitly understo od). In order to av oid trivial situations where particles hav e one offspring almost surely , we assume henceforth the following. F or all x ∈ X there is a vertex y ⇌ x such that µ y ( f : P w : w ⇌ y f ( w ) = 1) < 1, that is, in every equiv alence class (with resp ect to ⇌ ) there is at least one v ertex where a particle can hav e, inside the class, a num ber of c hildren different from 1 wpp. W e now introduce some definitions. The notation of the main quantities is summarized in T able 1. F or a more formal statement, see Definition 2.1 and Equations (3.2) and (3.3). Definition 2.1. Given a BR W ( X , µ ) , x ∈ X and A ⊆ X , the pr ob ability of extinction in A , starting with one p article at x , is define d as q ( x, A ) := 1 − P x (lim sup n →∞ X y ∈ A η n ( y ) > 0) . 3 Notation Meaning q ( x, A ) probabilit y of extinction in A , starting from x q ( A ) v ector whose comp onen ts are q ( · , A ) λ ( x, A ) threshold b etw een extinction and surviv al in A starting from x λ w ( x ) threshold b etw een global extinction and global surviv al, starting from x λ s ( x ) threshold b etw een extinction and surviv al in x , starting from x T able 1. Main notation. The BR W starts with one individual in x ∈ X . We denote by q ( A ) the extinction pr ob ability ve ctor, whose x -entry is q ( x, A ) . If A = { y } , we write q ( x, y ) instead of q ( x, { y } ). Note that q ( x, A ) dep ends on µ . When w e need to stress this dep endence, we write q µ ( x, A ). Extinction probabilities ha ve been the ob ject of in tense study during the last decades. W e refer the reader, for instance, to [4, 6, 34]. The ev ent S ( A ) := { lim sup n →∞ P y ∈ A η n ( y ) > 0 } is called survival in A , while E ( A ) := S ( A ) ∁ is called extinction in A . Definition 2.2. L et ( X, µ ) b e a BR W starting fr om one p article at x ∈ X and let A ⊆ X . We say that (1) the pr o c ess survives in A if q ( x, A ) < 1 ; (2) the pr o c ess survives globally if q ( x, X ) < 1 ; (3) ther e is strong surviv al in A if q ( x, A ) = q ( x, X ) < 1 and nonstrong surviv al in A if q ( x, X ) < q ( x, A ) < 1 ; (4) ther e is no strong surviv al in A if either q ( x, A ) = 1 or q ( x, X ) < q ( x, A ) ; (5) the pr o c ess is in a pure global surviv al phase if q ( x, X ) < q ( x, x ) = 1 . We say that ther e is (strong) lo cal surviv al/extinction in A when A is finite and nonempty. When there is no surviv al wpp, we say that there is extinction and the fact that extinction o ccurs almost surely , will b e tacitly understo o d. When there is strong surviv al in A , it means that for almost all realizations, the pro cess either surviv es in A (hence globally) or it go es globally extinct. More precisely , there is strong (lo cal) surviv al at y starting from x if and only if the probability of (lo cal) surviv al at y starting from x conditioned on global surviv al starting from x is 1. W e w ant to stress that, for a BR W starting from x ∈ X , q ( x, X ) = q ( x, A ) if and only if global surviv al is equiv alent to strong surviv al at A . On the other hand, q ( x, X ) < q ( x, A ) if and only if there is global surviv al and no strong surviv al at A . Henceforth, when we sa y lo c al survival without reference to any particular set A , w e mean lo c al survival in C for every finite non-empt y subset C ⊆ X . W e emphasize that when the pro cess is irreducible, q ( C ) do es not dep end on the choice of the finite non-empty C (see b elow). In general, Definition 2.2 dep ends on the starting vertex. How ever, if the pro cess is irreducible, then the pro cess surviv es in A or globally starting from x if and only if the same holds when starting from y , for all x, y ∈ X . Strong surviv al in A may still dep end on the starting vertex, ev en in the irreducible case. If at all sites we hav e a positive probability that a particle has no children, then strong surviv al do es not dep end on the starting vertex (see [7, Section 3]). Moreo ver, if a BR W is irreducible, A, B ⊂ X are nonempty , finite subsets and C ⊂ X is nonempty , then q ( x, A ) = q ( x, B ) ≥ q ( x, C ) for all x ∈ X . The first equality holds since lo cal surviv al do es not dep end on the target vertex and surviv al in a finite (nonempt y) set is equiv alent to local surviv al to a vertex in the set; more precisely , in the irreducible case, if a vertex y is visited infinitely often, then the conditional probabilit y of visiting infinitely man y times all other v ertices is 1 (b y Borel- Can telli Lemma). The second inequality follows from the fact that every nonempty set contains a finite nonempt y set. Analogously , in the irreducible case, q ( x, A ) < 1 for some x ∈ X if and only if q ( x, A ) < 1 for all x ∈ X . 4 Henceforth, we make use of the natural partial order b etw een vectors defined as follows: q ( A ) ≤ q ( B ) if and only if q ( x, A ) ≤ q ( x, B ) for all x ∈ X . Therefore, q ( A ) < q ( B ) if and only if q ( x, A ) ≤ q ( x, B ) for all x ∈ X and q ( x 0 , A ) < q ( x 0 , B ) for some x 0 ∈ X ; moreov er, q ( A ) ≰ q ( B ) if and only if q ( x 0 , A ) > q ( x 0 , B ) for some x 0 ∈ X . It is well-kno wn that extinction probability v ectors are fixed p oin ts of the generating function G µ : [0 , 1] X → [0 , 1] Y defined as follows: G µ ( z | y ) := X f ∈ S X µ y ( f ) Y x ∈ X z ( x ) f ( x ) , (2.1) where G µ ( z | y ) is the y -co ordinate of G µ ( z ). F or the prop erties of this generating function, see for instance [12, Section 2.1] and the references therein. 2.2. Equiv alen t sets and comparison of extinction probability v ectors. W e b egin by stating Theorem 2.3 (see [4, Theorem 4.1]), whic h is a fundamen tal to ol for comparing extinction probability v ectors. In the case of global surviv al (c ho ose B = X ), it giv es equiv alen t conditions for strong surviv al in terms of extinction probabilities. W e denote b y q 0 ( x, A ) the probability that the pro cess, whic h starts with one particle at x , has no progen y ever in the set A . Theorem 2.3 compares the extinction probabilities of tw o sets. Sp ecifically , it asserts that a BR W has a p ositiv e probability of surviving in a set B while becoming extinct in a set A if and only if it has a p ositiv e probabilit y of surviving in B without ever visiting A . The “if ” direction of this equiv alence is straightforw ard, whereas the conv erse is not. This is a very p ow erful result which has many applications. Theorem 2.3 ([4, Theorem 4.1]) . F or every BR W ( X , µ ) and A, B ⊆ X , the fol lowing statements ar e e quivalent: (1) ther e exists x ∈ X such that q ( x, B ) < q ( x, A ) ; (2) ther e exists x ∈ X such that q ( x, B \ A ) < q ( x, A ) ; (3) ther e exists x ∈ X such that q ( x, B ) < q 0 ( x, A ) ; (4) ther e exists x ∈ X \ A such that, starting fr om x ther e is a p ositive chanc e of survival in B without ever visiting A ; (5) ther e exists x ∈ X such that, starting fr om x ther e is a p ositive chanc e of survival in B and extinction in A ; (6) inf x ∈ X : q ( x,B ) < 1 1 − q ( x, A ) 1 − q ( x, B ) = 0 . W e define relations b et ween subsets A, B ⊆ X in a given BR W: w e write • A ⇒ B if surviv al in A implies surviv al in B from every starting p oin t (that is, P x ( S ( B ) ∩ S ( A )) = P ( S ( A )) for all x ∈ X ), • A ⇏ B if there is a p ositiv e chance of surviv al in A and extinction in B from some starting p oin ts (that is, P x ( S ( B ) ∩ S ( A )) < P ( S ( A )) for some x ∈ X ), • A ⇔ B if surviv al in A implies surviv al in B and vice-versa from every starting p oin t, • A ⇎ B if there is a p ositiv e chance of surviv al in B and extinction in A from some starting p oin ts and vice-v ersa. Note that A ⇔ A for all A ⊆ X . When necessary , we write µ ⇒ to emphasize the sp ecific BR W. The next corollary is a straightforw ard consequence of the equiv alence b etw een (i) and (v) in Theorem 2.3. Corollary 2.4 ([4, Corollary 5.1]) . L et A, B ⊆ X . (1) A ⇒ B if and only if q ( A ) ≥ q ( B ) . (2) A ⇔ B if and only if q ( A ) = q ( B ) . (3) A ⇎ B if and only if ther e is no or der r elation b etwe en q ( A ) and q ( B ) . 5 W e p oin t out that any of the six equiv alen t conditions in Theorem 2.3 can b e used to establish the relation b etw een the pair A, B ⊆ X . W e observ e that A ⇔ B is equiv alent to q ( A ) = q ( B ). Remark 2.5. If A 0 ⇒ A then A ⇔ A ∪ A 0 . Equivalently, q ( A 0 ) ≥ q ( A ) implies q ( A ) = q ( A ∪ A 0 ) . Inde e d, survival in A implies survival in A ∪ A 0 ; c onversely, survival in A ∪ A 0 either implies survival in A or survival in A 0 which, in turn, implies survival in A . Mor e over, if ( X, µ ) is irr e ducible then A ⇒ B for every nonempty A, B ⊆ X wher e A is finite (se e the discussion at the end of Se ction 2.1). Ther efor e, A ⇔ B for al l A, B ⊆ X nonempty and finite. 3. Germ-monotone f amilies of Branching Random W alks In the theory of BR Ws, a great num b er of pap ers hav e b een devoted to the contin uous-time v ersion of the pro cess. It is clear that, a contin uous-time BR W has a discrete-time counterpart defined as follows: then (random) num b er of children of a particle in discrete time is the total n umber of children of the particle in contin uous time in its whole lifetime (see Section 3.1 for more details). The tw o pro cesses hav e the same extinction probability , therefore, in a sense, a con tinuous- time pro cess is a particular case of a discrete-time pro cess. It will b e clear in the following that a con tinuous-time pro cess is, in fact, a family of pro cesses dep ending on a p ositiv e parameter λ > 0; for ev ery fixed λ there is a well defined contin uous-time BR W. Therefore, the discrete counterpart of a contin uous-time BR W is, in fact, a family of discrete-time pro cesses dep ending on a parameter λ . In this section we extend this with a more general definition of parameter dependent family of BR Ws (whose contin uous-time BR Ws is a particular case). T o this aim we need an idea introduced in [12]. Definition 3.1. L et µ := { µ x } x ∈ X and ν := { ν x } x ∈ X b e two families of me asur es on S X . L et G µ and G ν b e the asso ciate d gener ating functions. (1) µ ⪰ ν if and only if µ x ⪰ ν x for al l x ∈ X , that is, if and only if given a non-de cr e asing me asur able function F : S X → R , we have R F d µ x ≥ R F d ν x for al l x ∈ X such that the inte gr als ar e wel l define d. (2) µ ≥ p gf ν if and only if G µ ( z ) ≤ G ν ( z ) for al l z ∈ [0 , 1] X . (3) µ ≥ germ ν if and only if ther e exists δ ∈ [0 , 1) G µ ( z ) ≤ G ν ( z ) for al l z ∈ [ δ, 1] X . If # X = 1 , that is, µ = { µ } and ν = { ν } , then we simply write µ ≥ p gf ν and µ ≥ germ ν . W e observe that µ ⪰ ν ⇒ µ ≥ pgf ν ⇒ µ ≥ germ ν , but the rev erse implications do not hold. Clearly G µ ( z ) ≤ G ν ( z ) if and only if G µ ( z | y ) ≤ G ν ( z | x ) for all x ∈ X ; th us, µ ≥ germ ν (with a certain δ < 1 ) if and only if µ x ≥ germ ν x for all x ∈ X (with δ x suc h that sup x ∈ X δ x ≤ δ < 1). This leads to the following definition. Definition 3.2. Consider a family of me asur es M := { µ x,λ } x ∈ X,λ> 0 on S X such that if λ ≥ λ ∗ then µ λ ≥ germ µ λ ∗ (wher e µ λ := { µ x,λ } x ∈ X } ). The family of BR Ws { ( X, µ λ ) } λ> 0 is c al le d germ- monotone family of BR Ws or simply germ-monotone BR W (in short, GMBR W ) and denote d by ( X, M ) . Given λ > 0 , the λ -BR W asso ciate d to the GMBR W is denote d by ( X, µ λ ) . A GMBR W ( X , M ) is c al le d regular if and only if for al l x ∈ X the irr e ducible class [ x ] induc e d by ( X , µ λ ) do es not dep end on λ > 0 . In p articular, a GMBR W is c al le d irreducible if and only if ( X, µ λ ) is irr e ducible for al l λ > 0 (in this c ase it is obviously r e gular). Giv en a GMBR W ( X , M ) and a fixed λ > 0, we denote by q M ( A | λ ) (or q µ λ ( A )) the extinction probabilit y vector in A ⊆ X for the BR W ( X , µ λ ). When the family M is clear from the context, w e simply write q ( A | λ ). W e observe that λ 7→ q ( x, A | λ ) does not need to b e a monotone function in the most general case; ho wev er, if µ λ ≥ pgf µ λ ∗ (or µ λ ⪰ µ λ ∗ ) for all λ ≥ λ ∗ then λ 7→ q ( x, A | λ ) is nonincreasing (apply [12, Theorem 4.1(1)] with δ = 0). 6 Relations b etw een critical parameters and probabilities of extinction λ ≤ λ s ( x ) ⇐ ⇒ q ( x, { x }| λ ) = 1 λ > λ s ( x ) ⇐ ⇒ q ( x, { x }| λ ) < 1 λ < λ ( x, A ) = ⇒ q ( x, A | λ ) = 1, q ( x, A | λ ) = 1 = ⇒ λ ≤ λ ( x, A ) λ > λ ( x, A ) = ⇒ q ( x, A | λ ) < 1, q ( x, A | λ ) < 1 = ⇒ λ ≥ λ ( x, A ) T able 2. Relations b etw een critical parameters and extinction probabilities; see [5, Theorem 4.7 and Example 3] and [12, Theorem 4.1]. Giv en x ∈ X and A ⊆ X there is critical parameter λ M ( x, A ) asso ciated with the GMBR W, defined as λ M ( x, A ) := inf { λ > 0 : q M ( x, A | λ ) < 1 } . (3.2) When it is not imp ortan t to emphasize the GMBR W ( X , M ) then we simply write λ ( x, A ) instead of λ M ( x, A ). Among these parameters, t wo hav e been studied b y man y authors in the particular case of con tin uous-time BR Ws: the glob al survival critic al p ar ameter λ w ( x ) and the lo c al survival critic al p ar ameter λ s ( x ) (or, briefly , the global and lo cal critical parameters, resp ectiv ely) defined as λ w ( x ) := λ ( x, X ) , λ s ( x ) := λ ( x, { x } ) . (3.3) The relations betw een the critical parameters λ ( x, A ) and the extinction probabilities are sum- marized in T able 2 and they follo ws from [5, Theorem 4.7 and Example 3] and [12, Theorem 4.1]. In particular, the critical parameter is a threshold betw een a.s. extinction in A and surviv al in A with p ositive probability . Clearly A ⊆ B implies A ⇒ B whic h, in turns, implies λ ( x, A ) ≥ λ ( x, B ) for all x ∈ X . The smallest critical v alue is λ w , while the maximum b elongs to { λ s ( x ) } x ∈ X . Henceforth, unless otherwise explicitly mentioned, we consider only regular GMBR W. F or a reg- ular GMBR W these critical v alues dep end only on the irreducible class of x . In particular, they are constan t if the GMBR W is irreducible, in whic h case we simply write λ ( A ). Indeed, supp ose that x → y ; since there is a p ositiv e probability that a descendan t of a particle in x is placed in y , if there is a p ositiv e probability of surviv al in a set A starting from y , then the same holds starting from x . Therefore, λ ( x, A ) ≤ λ ( y , A ) and if x ⇌ y , then λ ( x, A ) = λ ( y , A ). Moreov er, if x is visited an infinite num b er of times, then, b y Borel-Can telli’s Lemma, an infinite num b er of descendants is placed at y : more precisely , the probabilit y of visiting y infinitely often conditioned on visiting x infinitely often is 1. Thus, if x ⇌ y then λ s ( x ) = λ s ( y ). If a GMBR W is irreducible, then λ ( A ) = λ s do es not dep end on the finite nonempt y subset A ⊆ X . This follo ws from the fact that q ( x, A | λ ) do es not dep end on the finite nonempty A . Moreov er, in this case, for every nonempty B ⊆ X we ha ve λ w ≤ λ ( B ) ≤ λ s . F or GMBR Ws the relation A ⇒ B introduced in Section 2.2 dep ends on λ ; if λ is not specified, it means that A µ λ ⇒ B for every λ > 0 and w e write A M ⇒ B . Similarly , A M ⇏ B means that there exists λ such that A µ λ ⇏ B . Giv en a GMBR W and a fixed λ > 0 then A µ λ ⇔ B is equiv alen t to q ( A | λ ) = q ( B | λ ); A ⇔ B is equiv alent to q ( A | λ ) = q ( B | λ ) for all λ > 0. A particular case of regular GMBR Ws is the discrete-time counterpart of a con tinuous-time BR W and, more generally , the discrete-time coun terpart of an aging BR W; in this case, λ ≥ λ ∗ implies µ λ ⪰ µ λ ∗ (see Section 3.1 for more details). 3.1. Con tinuous-time BR Ws and aging BR WS. W e start this section introducing the classic con tinuous-time BR W which is the simplest aging BR W; the general definition of aging BR W is the goal of the second part of this section. Giv en an at most countable X and a nonnegative matrix K = ( k xy ) x,y ∈ X , one can define a family of contin uous-time Branc hing Random W alks { η t } t ≥ 0 , where η t ( x ) represents the num b er of particles 7 aliv e at time t at site x , for any x ∈ X . The family is indexed by the repro ductiv e sp eed parameter λ > 0. Here is the dynamics. Each particle has an exp onen tially distributed lifetime with parameter 1. During its lifetime each particle aliv e at x breeds in to y according to the arriv al times of its o wn Poisson p oint pro cess with intensit y λk xy (represen ting the reproduction rate). W e denote by ( X, K ) this family of contin uous-time BR Ws (dep ending on λ > 0). It is not difficult to see that the in tro duction of a nonconstant death rate { d ( x ) } x ∈ X do es not represent a significant generalization. In fact, one can study a new BR W with death rate 1 and repro duction rates { λk xy /d ( x ) } x,y ∈ X ; the t wo pro cesses hav e the same b eha viors in terms of surviv al and extinction ([7, Remark 2.1]). T o eac h contin uous-time BR Ws w e associate a discrete-time coun terpart, namely ( X, µ ) where w e simply take in to account the n umber and the p ositions of all offspring b orn before the death of the parent. Clearly , from the knowledge of the discrete-time counterpart of ( X, K ), we cannot retriev e how many particles are aliv e at time t (and where), but the information ab out surviv al and extinction is in tact. The probabilities of extinction of a contin uous BR W coincide with those of its discrete-time coun terpart and they dep end on λ . In particular, we extend ev ery definition from the discrete-time case to the contin uous-time case in a natural wa y by using the discrete- time counterpart of a contin uous-time pro cess. More precisely , when w e say that a con tinuous-time pro cess has a certain property , w e mean that its discrete-time counterpart has it; for instance, a con tinuous-time process is irreducible (b y definition) if and only if its discrete-time counterpart is. It is easy to show that the exp ected num b er of children from x to y , for the discrete-time counterpart of the pro cess, is m xy = λk xy . The first moment matrix M equals λK . The discrete-time counterpart of a contin uous-time BR W is a regular GMBR W; more explicitly , for every fixed λ > 0, µ λ can b e written as µ x,λ ( f ) = ρ x,λ   X y ∈ X f ( y )   ( P y ∈ X f ( y ))! Q y ∈ X f ( y )! Y y ∈ X p ( x, y ) f ( y ) , ∀ f ∈ S X , ∀ x ∈ X , where ρ x,λ ( n ) :=  λ P y ∈ X k xy  n  1 + λ P y ∈ X k xy  n +1 , ∀ n ∈ N , p ( x, y ) := k xy P y ∈ X k xy , ∀ x, y ∈ X . In a more compact wa y µ x,λ ( f ) = 1  1 + λ P y ∈ X k xy  ( P y ∈ X f ( y ))! Q y ∈ X f ( y )! Y y ∈ X  λk xy 1 + λ P y ∈ X k xy  f ( y ) , (3.4) th us G µ λ ( z | x ) = 1 1 + λ P y ∈ X k xy (1 − z ( y )) for all z ∈ [0 , 1] X and x ∈ X . The regularity follo ws from the fact that the irreducible class of [ x ] is the usual irreducible class induced by the matrix K , which do es not dep end on λ > 0. Roughly sp eaking, in equation (3.4), ρ x,λ ( · ) is the la w of the total num b er of children of a particle living at x ∈ X , while the second part of the expression of µ x,λ tak es care of the indep endent random disp ersal of the children according to the diffusion law p ( x, · ). A more realistic con tinuous-time mo del is given b y the aging BR W , in which an individual’s repro ductiv e ability dep ends on its age (see, for instance, [13]). Let K := { κ xy } x,y ∈ X b e a family of nonnegativ e measurable functions such that κ xy ∈ L 1 loc ([0 , + ∞ )). F or each x ∈ X , we assume that the lifetime of a particle lo cated at x is gov erned b y a law with cumulativ e distribution function T x , where { T x } x ∈ X is a collection of nondecreasing, right-con tin uous functions satisfying T x (0) = 0 and lim t → + ∞ T x ( t ) = 1. The dynamics of the pro cess are as follo ws. When a particle is b orn at site x at time ¯ t , a family of indep enden t Poisson point process with intensities { λκ xy } y ∈ X is activ ated. During the random time in terv al [ ¯ t, ¯ t + ˆ t ], where ˆ t denotes the random lifetime of the particle with cum ulative distribution 8 function T x , at each arriv al time of a P oisson p oint pro cess with in tensity t 7→ λ κ xy ( t − ¯ t ) a new particle is placed at y . All Poisson p oint pro cesses and lifetimes, dep ending on x, y ∈ X and on the individual particle, are assumed to b e indep enden t. In general, this process is non-Mark ovian, unless the in tensities are constant and lifetimes are exp onentially distributed. Nonetheless, the associated discrete-time pro cess describing the num b er and locations of offspring at the end of an individual’s lifetime is a regular GMBR W. It is p ossible to explicitly compute the measures of the family µ λ as µ x,λ ( f ) = Z ∞ 0 Y y ∈ X  exp  − λ R t 0 κ xy ( s )d s  λ R t 0 κ xy ( s )d s  f ( y ) f ( y )!  P T x (d t ) (3.5) where P T x is the law whose c.d.f. is T x . If κ x,y = k x,y is a constant function and the lifetime is exp onen tially distributed with parameter 1 (i.e. T x ( t ) := 1 l [0 , + ∞ )] ( t )(1 − exp( − t )) then equation (3.5) b ecomes equation (3.4) and we hav e the usual contin uous-time BR W. This shows that dealing with GMBR Ws makes it p ossible to derive results for a broader class of pro cesses, extending b ey ond BR Ws and including non-Marko vian dynamics. Remark 3.3. We r e c al l that if a BR W ( X , µ ) is irr e ducible then q µ ( C ) do es not dep end on the choic e of a finite, nonempty C ⊆ X . Mor e over, if we ar e de aling with a c ontinuous-time BR W ( X , K ) , the map λ 7→ q ( A | λ ) is cle arly nonincr e asing, sinc e in this c ase µ λ ⪰ µ λ ∗ for al l λ ≥ λ ∗ > 0 . We observe that, even when the BR W ( X , K ) is homo gene ous, λ 7→ q ( A | λ ) do es not ne e d to b e c ontinuous in [0 , 1] X with the p ointwise c onver genc e top olo gy. An example is a quasi-tr ansitive BR W ( X , K ) wher e λ w < λ s such as, for instanc e, X = T d the homo gene ous tr e e with de gr e e d ≥ 3 wher e K is the adjiac ency matrix of the tr e e (se e [11, Corollary 3 and Figure 1] ). Inde e d, for any such BR W, ac c or ding to [7, Corollary 3.2] (se e also [12, Theorem 4.1] ) given any finite nonempty C ⊆ X , either q ( C | λ ) = 1 or q ( C | λ ) = q ( X | λ ) . Mor e over q ( X | λ ) < 1 for al l λ > λ w while q ( C | λ ) ( = 1 λ ≤ λ s < 1 λ > λ s ther efor e q ( C | λ ) = ( 1 λ ≤ λ s q ( X | λ ) λ > λ s . Sinc e q ( X | λ ) ≤ q ( X | λ s ) < 1 for al l λ ≥ λ s then lim λ → λ + s q ( C | λ ) ≤ q ( X | λ s ) < 1 = lim λ → λ − s q ( C | λ ) . 4. Critical p arameters for modified BR Ws and GMBR Ws In this section, w e examine how mo difications on a set may influence the b ehavior of a BR W. Since surviv al on a fixed set for a contin uous-time BR W is equiv alen t to surviv al on the same set for its discrete-time counterpart, it is natural to use results from the discrete-time case to infer corresp onding results for contin uous-time BR Ws. The main tec hnical to ol of this section is Theorem 2.3, from which we derive Theorem 4.1 and Prop osition 4.3. These results, in turn, provide insigh t into the b ehavior of a BR W under some mo difications. T o this end, let ( X, µ ) and ( X, ν ) be tw o BR Ws, and denote b y q µ and q ν their resp ectiv e extinction probability vectors. Similarly , we write µ ⇒ and ν ⇒ for the surviv al implications in tro duced in Section 2.2. The following result is a substantial generalization of [11, Theorem 3] and serv es as a cornerstone for the remainder of the pap er. Theorem 4.1. L et ( X , µ ) and ( X , ν ) b e two BR Ws and define A 0 := { x ∈ X : µ x  = ν x } . If A, B ⊆ X such that A 0 µ ⇒ A and A 0 ν ⇒ A , then q µ ( A ) ≤ q µ ( B ) ⇐ ⇒ q ν ( A ) ≤ q ν ( B ) . (4.6) 9 If, in addition, A 0 µ ⇒ B and A 0 ν ⇒ B , then      q µ ( A ) = q µ ( B ) ⇐ ⇒ q ν ( A ) = q ν ( B ) , q µ ( A ) < q µ ( B ) ⇐ ⇒ q ν ( A ) < q ν ( B ) , q µ ( A ) > q µ ( B ) ⇐ ⇒ q ν ( A ) > q ν ( B ); (4.7) mor e over q µ ( A ) and q µ ( B ) ar e not c omp ar able if and only if q ν ( A ) and q ν ( B ) ar e not c omp ar able. Final ly, if ( X, µ ) and ( X , ν ) ar e b oth irr e ducible and A 0 is finite, then e quation (4.7) holds for al l nonempty subsets A, B ⊆ X . Pr o of. According to Remark 2.5, q µ ( A ) = q µ ( A ∪ A 0 ) and q ν ( A ) = q ν ( A ∪ A 0 ), therefore it is enough to prov e equation (4.6) in the case A ⊇ A 0 and this has b een done in [11, Theorem 3]. This pro ves equation 4.6. W e now prov e the first line of equation (4.7), namely q µ ( A ) = q µ ( B ) ⇐ ⇒ q ν ( A ) = q ν ( B ) . (4.8) By Remark 2.5, if ( X, µ ) and ( X , ν ) are both irreducible and A 0 is finite then A 0 µ ⇒ A , A 0 µ ⇒ B , A 0 ν ⇒ A and A 0 ν ⇒ B for all nonempt y subsets A, B ; thus by applying equation (4.6) t wice (the second time by exchanging A and B ) then equation (4.8) follo ws. The remaining lines of equation (4.7) follow easily from equation (4.8) together with equation (4.6). Indeed, we ha ve pro ven that ( q µ ( A ) ≤ q µ ( B ) q µ ( A )  = q µ ( B ) ⇐ ⇒ ( q ν ( A ) ≤ q ν ( B ) q ν ( A )  = q ν ( B ) whic h is precisely the second line of equation (4.7). The third line follows by exchanging the role of A and B . Finally , from equation (4.7) it follo ws that q µ ( A ) and q µ ( B ) are not comparable if and only if q ν ( A ) and q ν ( B ) are not comparable. □ Remark 4.2. T o avoid unne c essary details in the statements of this se ction ’s r esults, we p oint out her e that, given two irr e ducible GMBR Ws ( X , M ) and ( X , N ) , if A 0 := { x ∈ X : µ x,λ  = ν x,λ for some λ > 0 } is finite, then Pr op ositions 4.3 and 4.6 and Cor ol lary 4.5 hold for every nonempty A, B ⊆ X . This c ase is discusse d in Se ction 5. Theorem 4.1, applied to a GMBR W, leads to Prop osition 4.3, whic h describ es how lo cal mo difi- cations of the GMBR W affect the critical parameters. This proposition extends [11, Corollary 2]. Note that min( λ M ( A ) , λ M ( B )) = max( λ M ( A ) , λ M ( B )) if and only if λ M ( A )  = λ M ( B ). Henceforth, giv en a, b ∈ R , we use the notation a ∨ b and a ∧ b for min( a, b ) and max( a, b ). Prop osition 4.3. L et ( X, M ) and ( X , N ) b e two irr e ducible GMBR Ws and A 0 := { x ∈ X : µ x,λ  = ν x,λ for some λ > 0 } . Supp ose that A 0 M ⇒ A , A 0 M ⇒ B , A 0 N ⇒ A and A 0 N ⇒ B . Then the fol lowing ar e e quivalent: (1) λ M ( A ) ∧ λ M ( B ) < λ N ( A ) ∧ λ N ( B ) ; (2) λ M ( A ) ∨ λ M ( B ) < λ N ( A ) ∧ λ N ( B ) ; (3) λ M ( A ) = λ M ( B ) < λ N ( A ) ∧ λ N ( B ) . Mor e over, if λ M ( A )  = λ M ( B ) then we have: (4) λ N ( A ) ∧ λ N ( B ) ≤ λ M ( A ) ∧ λ M ( B ) ; (5) if, in addition, λ N ( A )  = λ N ( B ) then λ N ( A ) ∧ λ N ( B ) = λ M ( A ) ∧ λ M ( B ) and λ M ( A ) = λ M ( A ) ∧ λ M ( B ) ⇐ ⇒ λ N ( A ) = λ N ( A ) ∧ λ N ( B ) , λ M ( B ) = λ M ( A ) ∧ λ M ( B ) ⇐ ⇒ λ N ( B ) = λ N ( A ) ∧ λ N ( B )) (4.9) (6) either λ N ( A ) = λ N ( B ) ≤ λ M ( A ) ∧ λ M ( B ) or λ N ( A ) ∧ λ N ( B ) = λ M ( A ) ∧ λ M ( B )) < λ N ( A ) ∨ λ N ( B ) . 10 Pr o of. According to Remark 2.5, q M ( A | λ ) = q M ( A ∪ A 0 | λ ), q N ( A | λ ) = q N ( A ∪ A 0 | λ ), q M ( B | λ ) = q M ( B ∪ A 0 | λ ) and q N ( B | λ ) = q N ( B ∪ A 0 | λ ) for all λ > 0 therefore λ M ( A ) = λ M ( A ∪ A 0 ), λ N ( A ) = λ N ( A ∪ A 0 ), λ M ( B ) = λ M ( B ∪ A 0 ) and λ N ( B ) = λ N ( B ∪ A 0 ). Hence, it is enough to prov e the statemen t under the assumption A 0 ⊆ A, B . Clearly (3) = ⇒ (2) = ⇒ (1). W e prov e (1) = ⇒ (3). T o this aim, supp ose that λ M ( A ) = λ M ( A ) ∧ λ M ( B )) < λ N ( A ) ∧ λ N ( B ). Consider λ ∈  λ M ( A ) , λ N ( A ) ∧ λ N ( B )  . In this case q N ( A | λ ) = q N ( B | λ ) = 1 ; th us, applying Theorem 4.1, w e obtain q M ( A | λ ) = q M ( B | λ ) < 1 . Since q M ( B | λ ) < 1 for all λ > λ M ( A ) we hav e λ M ( B ) ≤ λ M ( A ) < λ N ( A ) ∧ λ N ( B ). Since λ M ( A ) = λ M ( A ) ∧ λ M ( B ) we hav e λ M ( A ) = λ M ( B ). The case λ M ( B ) = λ M ( A ) ∧ λ M ( B ) is completely analogous: just exchange the role of A and B . W e now prov e (4). By con tradiction, if λ N ( A ) ∧ λ N ( B ) > λ M ( A ) ∧ λ M ( B ) then b y the equiv alence of (1) and (3) we hav e λ M ( A ) = λ M ( B ) which is a contradiction. W e prov e (5). F rom (2) we ha ve λ M ( A )  = λ M ( B ) = ⇒ λ N ( A ) ∧ λ N ( B ) ≤ λ M ( A ) ∧ λ M ( B ) λ N ( A )  = λ N ( B ) = ⇒ λ M ( A ) ∧ λ M ( B ) ≤ λ N ( A ) ∧ λ N ( B ) therefore λ N ( A ) ∧ λ N ( B ) = λ M ( A ) ∧ λ M ( B ). Now, since λ M ( A )  = λ M ( B ) and λ N ( A )  = λ N ( B ) then the tw o double implications in equation‘4.9 are equiv alen t; thus, let us prov e the first line. Supp ose now, by contradiction, that λ 0 := λ M ( A ) = λ M ( A ) ∧ λ M ( B ) = λ N ( A ) ∧ λ N ( B ) = λ N ( B ) < λ M ( B ) ∧ λ N ( A ). T ake λ ∈  λ 0 , λ M ( B ) ∧ λ N ( A )  ; we hav e q M ( A ∪ A 0 | λ ) = q M ( A | λ ) < 1 = q M ( B | λ ) = q M ( B ∪ A 0 | λ ) q N ( B ∪ A 0 | λ ) = q N ( B | λ ) < 1 = q N ( A | λ ) = q N ( A ∪ A 0 | λ ) whic h contradicts Theorem 4.1. This shows that λ M ( A ) = λ M ( A ) ∧ λ M ( B ) implies λ N ( A ) = λ N ( A ) ∧ λ N ( B ). The conv erse is prov en by switching the role b etw een λ M and λ N . W e prov e (6). According to (4), if λ N ( A ) = λ N ( B ) ≤ λ M ( A ) ∧ λ M ( B ) do es not hold, then λ N ( A ) ∧ λ N ( B ) < λ N ( A ) ∨ λ N ( B ), that is, λ N ( A )  = λ N ( B ); thus, according to (5) λ N ( A ) ∧ λ N ( B ) = λ M ( A ) ∧ λ M ( B ). □ Note that in the previous prop osition (3) can b e equiv alen tly written as λ M ( A ) ∧ λ M ( B ) = λ M ( A ) ∨ λ M ( B ) < λ N ( A ) ∧ λ N ( B ). The following corollary essentially gives an equiv alent condition for λ M ( A ) = λ M ( B ). Indeed, under the conditions of the corollary for A, B ⊆ X , if λ N ( A )  = λ N ( B ) then λ M ( A ) = λ M ( B ) is equiv alen t to λ M ( A ) ∨ λ M ( B ) ≤ λ N ( A ) ∧ λ N ( B ). Corollary 4.4. L et ( X, M ) and ( X , N ) b e two irr e ducible GMBR Ws and A 0 := { x ∈ X : µ x,λ  = ν x,λ for some λ > 0 } . If A 0 M ⇒ A , A 0 M ⇒ B , A 0 N ⇒ A and A 0 N ⇒ B then we have (1) λ M ( A ) ∨ λ M ( B ) ≤ λ N ( A ) ∧ λ N ( B ) implies λ M ( A ) = λ M ( B ) ≤ λ N ( A ) ∧ λ N ( B ) ; (2) if, in addition, λ N ( A )  = λ N ( B ) then λ M ( A ) ∨ λ M ( B ) > λ N ( A ) ∧ λ N ( B ) implies λ M ( A )  = λ M ( B ) and λ M ( A ) ∨ λ M ( B ) > λ M ( A ) ∧ λ M ( B ) = λ N ( A ) ∧ λ N ( B ) . Pr o of. Supp ose that λ M ( A ) ∨ λ M ( B ) ≤ λ N ( A ) ∧ λ N ( B ); in this case w e hav e either λ M ( A ) ∧ λ M ( B ) = λ M ( A ) ∨ λ M ( B ) (thus λ M ( A ) = λ M ( B )) or else λ M ( A ) ∧ λ M ( B ) < λ M ( A ) ∨ λ M ( B ) which implies λ M ( A ) ∧ λ M ( B ) < λ N ( A ) ∧ λ N ( B ) (thus, again, λ M ( A ) = λ M ( B )). Con versely , supp ose that λ N ( A )  = λ N ( B ) and λ M ( A ) ∨ λ M ( B ) > λ N ( A ) ∧ λ N ( B ). By exchanging the role of M and N in Prop osition 4.3(4) we hav e λ N ( A ) ∧ λ N ( B ) ≥ λ M ( A ) ∧ λ M ( B ); therefore λ M ( A ) ∨ λ M ( B ) > λ M ( A ) ∧ λ M ( B ), thus λ M ( A )  = λ M ( B ) and, b y Proposition 4.3(5), λ M ( A ) ∧ λ M ( B ) = λ M ( A ) ∧ λ M ( B ). The last tw o equations yield the claim. □ When ( X , M ) and ( X , N ) are t wo irreducible GMBR Ws and A and B satisfy the conditions of Prop osition 4.3 (if A 0 is finite then every A and B will do), then the only alternatives are: (1) λ M ( A ) = λ M ( B ) < λ N ( A ) ∧ λ N ( B ), (2) λ M ( A ) = λ M ( B ) = λ N ( A ) ∧ λ N ( B ), 11 (3) λ M ( A ) = λ N ( A ) = λ N ( B ) < λ M ( B ) (4) λ M ( A ) = λ N ( A ) < λ M ( B ) ∧ λ N ( B ), and those obtained by switching A and B or by switching M and N . Corollary 4.5. L et ( X, M ) and ( X , N ) b e two irr e ducible GMBR Ws and A 0 := { x ∈ X : µ x,λ  = ν x,λ for some λ > 0 } . Supp ose that A 0 M ⇒ A , A 0 M ⇒ B , A 0 N ⇒ A and A 0 N ⇒ B . Then we have (1) either λ N ( A ) ∧ λ N ( B ) ≤ λ M ( A ) ∧ λ M ( B ) or λ M ( A ) = λ M ( B ) , (2) if λ M ( B ) > λ M ( A ) then λ M ( A ) ≥ λ N ( A ) ; e quivalently, either λ M ( A ) ≥ λ N ( A ) or λ M ( B ) ≤ λ M ( A ) . Pr o of. (1) Observ e that if λ N ( A ) ∧ λ N ( B ) ≤ λ M ( A ) ∧ λ M ( B ) fails then according to Proposition 4.3 w e hav e λ M ( A ) = λ M ( B ). (2) If λ M ( A ) < λ M ( B ) and, according to Prop osition 4.3(3) and (4) we hav e λ M ( A ) ∧ λ M ( B ) ≥ λ N ( A ) ∧ λ N ( B ) and λ N ( A ) ≤ λ N ( B ) resp ectiv ely . Therefore λ M ( A ) = λ M ( A ) ∧ λ M ( B ) ≥ λ N ( A ) ∧ λ N ( B ) = λ N ( A ). □ This corollary leads to the following significant conclusion (see also Corollary 5.4): giv en a nonempt y A ⊆ X , either λ ( A ) attains its maxim um v alue (under finite mo difications, that is, if A 0 is finite), or λ ( B ) ≤ λ ( A ) for all nonempty subsets B ⊆ X , that is, λ ( A ) = λ s (since λ ( B ) ≤ λ s for all nonempt y subsets B ⊆ X and equality is attained when B is finite). F or results on finite mo difications see Section 5. The previous results deal with the minimum b etw een λ ( A ) and λ ( B ). The following prop osition, on the other hand, is ab out the maximum betw een λ ( A ) and λ ( B ). In the particular case where A is finite and nonempty and B := X , the previous results deal with λ w , while the following prop osition deals with λ s (see also Section 5.1). Prop osition 4.6. L et ( X, M ) and ( X , N ) b e two irr e ducible GMBR Ws and A 0 := { x ∈ X : µ x,λ  = ν x,λ for some λ > 0 } . Consider A, B ⊆ X such that A 0 M ⇒ A , A 0 M ⇒ B , A 0 N ⇒ A and A 0 N ⇒ B . If λ M ( A ) ∨ λ N ( B ) < λ M ( B ) then for al l λ ∈  λ M ( A ) ∨ λ N ( B ) , λ M ( B )  for the ( X , ν λ ) -BR W we have P x ( S ( B )) > 0 and P x ( S ( A ) ∩ E ( B )) > 0 for al l x ∈ X . In p articular, if B is nonempty and finite, A = X and λ M w ∨ λ N s < λ M s then for al l λ ∈  λ M w ∨ λ N s , λ M s  the ( X , ν λ ) -BR W has nonstr ong lo c al survival starting fr om al l x ∈ X . Pr o of. W e start by pro ving the first part of the prop osition. If λ ∈  λ M ( A ) , λ M ( B )  w e ha ve q M ( B | λ ) = 1 > q M ( A | λ ). According to Theorem 4.1, 1 > q N ( B | λ ) > q N ( A | λ ) where the first inequalit y follows from λ > λ N ( B ). Theorem 2.3 yields the conclusion. The second part follows easily since λ M ( A ) = λ M w , λ M ( B ) = λ M s , λ N ( A ) = λ N w and λ N ( B ) = λ N s . The fact that surviv al do es not depend on the starting point comes from the fact that, in the irreducible case, q ( x, A ) < 1 for some x ∈ X if and only if q ( x, A ) < 1 for all x ∈ X . □ 5. Critical p arameters for GMBR Ws under finite modifica tions Giv en the set X , let us consider the set X := { M : ( X , M ) is irreducible } of irreducible GMBR Ws on X ; note that a GMBR W is uniquely iden tified by its family M . Define the relation R on X by M R N if and only if ∆ M , N := { x ∈ X : µ x,λ  = ν x,λ , for some λ > 0 } is finite. The relation R is reflexiv e (since ∆ M , M = ∅ for all M ∈ X ), symmetric (since ∆ M , N = ∆ N , M for all M , N ∈ X ) and transitiv e (since ∆ M , N ⊆ ∆ M , I ∪ ∆ I , N for all M , I , N ∈ X ). Therefore R is an equiv alence relation; w e denote by [ M ] the equiv alence class of M and by X / R the quotient set. As a consequence of Remark 2.5, let ( X, M ) b e an irreducible GMBR W and let A, B ⊆ X b e nonempt y subsets, with A finite. Then A M ⇒ B , that is, A µ λ ⇒ B for all λ > 0. W e can therefore restate the main results of Section 4 in this imp ortan t sp ecial case. The crucial observ ation is that, 12 if M R N , then ∆ M , N ⇒ A for every nonempty subset A ⊆ X ; the set ∆ M , N th us pla ys the same role as the set A 0 in the statements of Section 4. Theorem 5.1. L et ( X , M ) and ( X , N ) b e two irr e ducible GMBR Ws such that M R N ; then for al l nonempty subsets A, B ⊆ X and for al l λ > 0 we have      q M ( A | λ ) = q M ( B | λ ) ⇐ ⇒ q N ( A | λ ) = q N ( B | λ ) , q M ( A | λ ) < q M ( B | λ ) ⇐ ⇒ q N ( A | λ ) < q N ( B | λ ) , q M ( A | λ ) > q M ( B | λ ) ⇐ ⇒ q N ( A | λ ) > q N ( B | λ ); mor e over q M ( A | λ ) and q M ( B | λ ) ar e not c omp ar able if and only if q N ( A | λ ) and q N ( B | λ ) ar e not c omp ar able. Prop osition 5.2. L et ( X, M ) and ( X, N ) b e two irr e ducible GMBR Ws such that M R N ; then for al l nonempty subsets A, B ⊆ X then the fol lowing ar e e quivalent: (1) λ M ( A ) ∧ λ M ( B ) < λ N ( A ) ∧ λ N ( B ) ; (2) λ M ( A ) ∨ λ M ( B ) < λ N ( A ) ∧ λ N ( B ) ; (3) λ M ( A ) = λ M ( B ) < λ N ( A ) ∧ λ N ( B ) . Mor e over, if λ M ( A )  = λ M ( B ) then we have: (4) λ N ( A ) ∧ λ N ( B ) ≤ λ M ( A ) ∧ λ M ( B ) ; (5) if, in addition, λ N ( A )  = λ N ( B ) then λ N ( A ) ∧ λ N ( B ) = λ M ( A ) ∧ λ M ( B ) and λ M ( A ) = λ M ( A ) ∧ λ M ( B ) ⇐ ⇒ λ N ( A ) = λ N ( A ) ∧ λ N ( B ) , λ M ( B ) = λ M ( A ) ∧ λ M ( B ) ⇐ ⇒ λ N ( B ) = λ N ( A ) ∧ λ N ( B ) . (6) either λ N ( A ) = λ N ( B ) ≤ λ M ( A ) ∧ λ M ( B ) or λ N ( A ) ∧ λ N ( B ) = λ M ( A ) ∧ λ M ( B )) < λ N ( A ) ∨ λ N ( B ) . Corollary 5.3. L et ( X , M ) and ( X, N ) b e two irr e ducible GMBR Ws such that M R N ; then for al l nonempty subsets A, B ⊆ X we have (1) λ M ( A ) ∨ λ M ( B ) ≤ λ N ( A ) ∧ λ N ( B ) implies λ M ( A ) = λ M ( B ) ≤ λ N ( A ) ∧ λ N ( B ) ; (2) if, in addition, λ N ( A )  = λ N ( B ) then λ M ( A ) ∨ λ M ( B ) > λ N ( A ) ∧ λ N ( B ) implies λ M ( A )  = λ M ( B ) and λ M ( A ) ∨ λ M ( B ) > λ M ( A ) ∧ λ M ( B ) = λ N ( A ) ∧ λ N ( B ) . Corollary 5.4. L et ( X , M ) and ( X, N ) b e two irr e ducible GMBR Ws such that M R N ; then for al l nonempty subsets A, B ⊆ X we have (1) either λ N ( A ) ∧ λ N ( B ) ≤ λ M ( A ) ∧ λ M ( B ) or λ M ( A ) = λ M ( B ) , (2) if, in addition, λ M ( B ) > λ M ( A ) then λ M ( A ) ≥ λ N ( A ) . In p articular, for every M and every nonempty A ⊆ X , either λ M ( A ) ≥ λ N ( A ) for every N ∈ [ M ] or λ M ( A ) ≥ λ M ( B ) for every nonempty B ⊆ X (or, e quivalently, λ M ( A ) = λ M s ). Pr o of. The result follows from Corollary 4.5. The additional remark, namely the equiv alence b etw een λ M ( A ) ≥ λ M ( B ) for every nonempt y B ⊆ X and λ M ( A ) = λ M s , can b e pro ved as follo ws: clearly for ev ery nonempt y B we ha ve λ M ( B ) ≤ λ M s ; conv ersely , if λ M ( A ) ≥ λ M ( B ) for every nonempty B ⊆ X then, by taking B finite, we hav e λ M ( A ) ≥ λ M s (while λ M ( A ) ≤ λ M s since A is nonempty). □ Since λ s is the maximal critical v alue, it pla ys a particularly imp ortan t role; therefore the follo wing result is relev ant. Corollary 5.5. L et ( X , M ) and ( X , N ) b e two irr e ducible GMBR Ws such that M R N and c onsider a subset A ⊆ X . If λ M ( A ) < λ M s then either λ N s = λ N ( A ) ≤ λ M ( A ) or λ N ( A ) = λ M ( A ) < λ N s . Pr o of. Note that in this case λ ( A ) ∧ λ s = λ ( A ) and λ ( A ) ∨ λ s = λ s . The claim follo ws from Prop osition 5.2(6). Alternativ ely , b y Corollary 5.4(2) we ha ve λ M ( A ) ≥ λ N ( A ); therefore if λ N s = λ N ( A ) ≤ λ M ( A ) do es not hold then we hav e necessarily λ N s  = λ N ( A ). Moreov er, if λ N s  = λ N ( A ) 13 then λ N s > λ N ( A ) thus, again according to Corollary 5.4(2) we ha ve λ M ( A ) ≤ λ N ( A ), whence λ M ( A ) = λ N ( A ) < λ N s . □ Prop osition 5.6. L et ( X, M ) and ( X , N ) b e two irr e ducible GMBR Ws such that M R N and fix two nonempty subsets A, B ⊆ X such that λ M ( A ) ∨ λ N ( B ) < λ M ( B ) . Then, for al l λ ∈  λ M ( A ) ∨ λ N ( B ) , λ M ( B )  , for the ( X , ν λ ) -BR W we have P x ( S ( B )) > 0 and P x ( S ( A ) ∩ E ( B )) > 0 for al l x ∈ X . An application of these results is given in the follo wing section. 5.1. Classic critical parameters for GMBR Ws and contin uous-time BR Ws under finite mo difications. Here w e sp ecify our general results for irreducible GMBR Ws by considering the case of the usual critical parameters λ w and λ s and finite mo difications. The following results can b e stated, for instance, for the sub class of discrete-time counterpart of contin uous-time BR Ws as in [11] (see Remark 5.11 for details). The first result follows from Prop osition 5.2 (see also [11, Corollary 2]). Corollary 5.7. L et ( X , M ) and ( X, N ) b e two irr e ducible GMBR Ws such that M R N . The fol lowing ar e e quivalent: (1) λ M w < λ N w , (2) λ M s < λ N w , (3) λ M w = λ M s < λ N w . Final ly, if λ M s ≤ λ N w then λ M w = λ M s ≤ λ N w ; if, in addition, λ N w < λ N s then λ M s ≥ λ N w implies λ M s > λ M w = λ N w . The second result deals with the maximality of the global critical parameter λ w . If a class con tains at least one GMBR W with pure global surviv al phase, then each and every one of them has the same parameter λ w whic h attains the maximum v alue inside the class. Roughly sp eaking, either λ w attains the maxim um v alue or λ w = λ s . it was originally prov en for contin uous-time BR Ws in [11, Proposition 2] but no w it follows from Proposition 5.2 and Corollary 5.4 by choosing a finite nonempt y A and B := X . Corollary 5.8. L et ( X, M ) such that λ M w < λ M s . Then for al l N ∈ [ M ] we have λ N w ≤ λ M w . Mor e over, for al l N ∈ [ M ] such that λ N w < λ N s , we have λ N w = λ M w . See [11] for a full review on the global critical v alue in a class. The second result of this section deals with the maximality of the lo cal critical parameter. here- after, by lo c al survival phase w e mean survival in a finite nonempty set (i.e. surviv al in every finite nonempt y set since the pro cess is irreducible). In this case, if a class contains at least one GM- BR W with pure global surviv al phase and no nonstrong local surviv al phase, then eac h and ev ery one of them has the same global and lo cal critical parameters whic h attain the maxim um v alue in the class. It follo ws from Proposition 5.10 since λ ( A ) = λ s for every finite nonempty A ⊆ X and ev ery BR W. This result can b e stated, in particular, for the sub class of discrete-time counterparts of contin uous-time BR Ws as describ ed in Remark 5.11. Corollary 5.9. L et ( X , M ) and ( X , N ) b e two irr e ducible GMBR Ws such that M R N . Supp ose that λ N s ∨ λ M w < λ ≤ λ M s , then ( X , N ) has a nonstr ong survival phase in every nonempty, finite set. These results imply the following. Prop osition 5.10. L et ( X, M ) b e an irr e ducible GMBR W. (1) If λ M w < λ M s then, for al l N ∈ [ M ] , either λ N w = λ M w , λ N s ≥ λ M s or ( X , N ) exhibits a nonstr ong lo c al survival phase. 14 (2) If λ M w < λ M s and ( X , M ) has no nonstr ong lo c al survival phase, then for al l N ∈ [ M ] we have λ M s ≥ λ N s . Mor e over, for al l N ∈ [ M ] such that ( X, N ) has no nonstr ong lo c al survival phase, we have λ N s = λ M s and λ N w = λ M w . Pr o of. (1) The first part comes from Corollary 5.9. Indeed, since λ M w < λ M s , and ( X, N ) do es not ha ve nonstrong surviv al phase, then λ N s ∨ λ M w ≥ λ M s , that is λ N s ≥ λ M s . Moreo ver, from Corollary 5.8, λ N w ≤ λ M w , whence λ N w < λ N s ; thus, again by Corollary 5.8, λ N w = λ M w . (2) W e s tart b y proving that λ M s ≥ λ N s . Since λ M w < λ M s then, according to Corollary 5.8, either λ M w = λ N w or λ N s = λ N w < λ M w . In the second case the assertion is prov ed. In the first case, we can apply (1) b y switc hing the role b et ween M and N : λ N w < λ M w , ( X , M ) does not exhibit a nonstrong surviv al phase therefore λ M s ≥ λ N s . W e now prov e the second assertion of (2). W e already prov ed that λ M s ≥ λ N s . Now, since λ M w < λ M s and ( X , N ) do es not hav e a nonstrong lo cal surviv al phase, then from (1) we hav e λ N w = λ M w , λ N s ≥ λ M s . Therefore, λ N w = λ M w and λ N s = λ M s . □ Remark 5.11. The r esults of Se ctions 4 and 5 apply natur al ly to a sub class of GOBR Ws, namely the discr ete-time c ounterp arts of c ontinuous-time BR Ws (and those arising fr om aging BR Ws). Inde e d, a c ontinuous-time BR W ( X, K ) is uniquely determine d by the matrix K = ( k xy ) x,y ∈ X . L et e X := { K : K = ( k xy ) x,y ∈ X is irr e ducible } and define a r elation f R on e X by setting K f R K ∗ if and only if the set ∆ K,K ∗ := { x ∈ X : ∃ y ∈ X, k xy  = k ∗ xy } is finite. This is pr e cisely the fr amework c onsider e d in [11] . The discr ete-time c ounterp art of a c ontinuous-time BR W intr o duc e d in Se ction 3.1 is a GMBR W (or der e d with r esp e ct to the sto chastic or der ⪰ , and henc e germ-or der e d as wel l). The c ontinuous- time pr o c ess and its discr ete-time c ounterp art shar e the same extinction pr ob ability ve ctors and critic al p ar ameters; mor e over, they have the same irr e ducible classes, and the r elation f R b etwe en c ontinuous-time BR Ws c orr esp onds to the r elation R (define d in Se ction 5) b etwe en their discr ete- time c ounterp arts. Conse quently, the r esults of Se ctions 4 and 5 c an b e tr ansferr e d verb atim to c ontinuous-time BR Ws by r eplacing R with f R . In this sense, our r esults signific antly extend those obtaine d in [11] . As an application of our results to the sub class of con tin uous-time BR Ws (as explained in the previous remark), we can pro ve the follo wing. W e recall that ( X , K ) is quasitr ansitive if a finite X 0 ⊂ X exists such that for every x ∈ X , there is a bijective map γ x : X → X s atisfying γ − 1 x ( x ) ∈ X 0 and k y z = k γ x y γ x z for all y, z . F or example, if the rates are translation-in v ariant, then ( X , K ) is quasi-transitiv e (actually , it is transitive). Corollary 5.12. (1) An irr e ducible c ontinuous-time BR W with pur e glob al survival phase and an irr e ducible c ontinuous-time BR W with no pur e glob al survival phase and no nonstr ong lo c al survival phase c annot b e in the same e quivalenc e class with r esp e ct to e R . (2) An irr e ducible c ontinuous-time BR W with pur e glob al survival phase and an irr e ducible qua- sitr ansitive c ontinuous-time BR W with no pur e glob al survival phase c annot b e in the same e quivalenc e class with r esp e ct to e R . Pr o of. (1) W e pro ve the result b y con tradiction. Let ( X, K ) b e suc h that λ K w < λ K s and ( X , K ∗ ) b e suc h that λ K ∗ w = λ K ∗ s and it do es not hav e a nonstrong lo cal surviv al phase. If K f R K ∗ then, according to Prop osition 5.10(1) λ K ∗ w = λ K w < λ K s ≤ λ K ∗ s and this contradicts λ K ∗ w = λ K ∗ s . Therefore, the tw o pro cesses are not in the same equiv alence class. 15 (2) It follo ws from (1) if we prov e that a quasitransitive BR W has no nonstrong lo cal surviv al phase. This is a consequence of [7, Corollary 3.2]; indeed, for all x ∈ X either q ( x, x ) = 1 or q ( x, x ) = q ( x, X ). Since the BR W is irreducible, then either q ( x, x ) = 1 for all x ∈ X or q ( x, x ) = q ( x, X ) for all x ∈ X . This yields the claim. □ 6. An example: the tree T d ⊕ T q In this section, we provide an example sho wing that λ w < λ ( A ) < λ s is p ossible for some A ⊆ X and for irreducible GMBR Ws on X and we study how these critical v alues are affected by a sp ecific lo cal mo dification. The pro cess, in this case, is the discrete-time counterpart of a con tinuous-time BR W. Let d, q ∈ N be suc h that d ≥ 6 and 2 √ d − 1 < q < d ; clearly q ≥ 5. Denote b y X the v ertex set of the graph T d ⊕ T q obtained by iden tifying the ro ots of T d and T q , while the set of edges is the union of the natural sets of edges of the trees T q and T d . Roughly sp eaking, w e are gluing the trees by the ro ots; we denote the new ro ot by o . Let K be the adjacency matrix of the graph: the ro ot has d + q neigh b ors, while the n umber of neigh b ors of x is q (resp. q ) if x is not the ro ot and b elongs to the subset T d (resp. T q ). In order to compute λ s , we take adv antage of the following characterization (see [5] or [34], see also [11]): λ s ( x ) = max { λ ∈ R : Φ( x, x | λ ) ≤ 1 } = sup { λ ∈ R : Φ( x, x | λ ) < 1 } (6.10) where Φ( x, y | λ ) := ∞ X n =1 φ ( n ) xy λ n and φ ( n ) xy := X x 1 ,...,x n − 1 ∈ X \{ y } k xx 1 k x 1 x 2 · · · k x n − 1 y . Roughly sp eaking, λ n φ ( n ) xy is the exp ected num ber of particles alive at y at time n , when the initial state is just one particle at x and the process b eha ves lik e a BR W except that ev ery particle reaching y at an y time i < n is immediately killed (b efore breeding). It is v ery easy to pro ve that, since the remo v al of o disconnects T d and T q then Φ X ( o, o | λ ) = Φ T d ( o, o | λ ) + Φ T q ( o, o | λ ) where Φ T n ( o, o | λ ) = n  1 − p 1 − 4( n − 1) λ 2  / (2( n − 1)) for all n ∈ N , n ≥ 2. W e now prov e that λ w = 1 /d , λ s = 1 / (2 √ d − 1), λ ( T q ) = q and λ ( T d ) = 1 /d . (1) λ s = 1 / (2 √ d − 1). Indeed, by equation (6.10) λ s ( x ) = sup n λ ∈ R : q 1 − p 1 − 4( q − 1) λ 2 2( q − 1) + d 1 − p 1 − 4( d − 1) λ 2 2( d − 1) < 1 o where d ≥ 6. Clearly λ s ≤ 1 / (2 √ d − 1) whic h is the radius of con vergence of the pow er series Φ( x, x | λ ). On the other hand, Φ X  o, o | 1 / (2 √ d − 1)  = q 1 − p 1 − ( q − 1) / ( d − 1) 2( q − 1) + d 2( d − 1) < 1 where the last inequality holds since d > q ≥ 5 (note that, for every fixed q ∈ N , q ≥ 2, the function d 7→ q 1 − √ 1 − ( q − 1) / ( d − 1) 2( q − 1) + d 2( d − 1) is strictly decreasing in [ q , + ∞ )). More precisely , the R-H.S. of the previous equation is bounded from ab ov e (for all q , d such that q < d ) b y the same expression where d = q + 1, that is Φ X  o, o | 1 / (2 √ d − 1)  ≤ q 1 − √ 1 /q 2( q − 1) + q +1 2 q = 1 − 1 2( √ q +1) + 1 q . It is easy to show that 1 − 1 2( √ q +1) + 1 q < 1 for all q ≥ 5. (2) λ w = 1 /d . Indeed, if λ ≤ 1 /d then there is no surviv al if the pro cess is restricted to T d or T q , thus there is extinction in X . Indeed, if there were surviv al for λ < 1 /d , it would b e pure global surviv al since λ < λ s = 1 / (2 √ d − 1). Therefore, according to Theorem 2.3, there is 16 a p ositiv e probability of surviving starting from some x ∈ X without visiting o . This would imply that the pro cess restricted to T d or to T q surviv es, but this is false. Con versely , if λ > 1 /d there is surviv al even if the pro cess is restricted to T d . Thus λ s ≥ 1 / (2 √ d − 1); hence λ s = 1 / (2 √ d − 1). (3) λ ( T q ) = 1 /q . Clearly λ ( T q ) ≤ 1 /q since if λ > 1 /q the pro cess restricted to T q surviv es. If there w ere surviv al for λ < 1 /q it w ould be pure global surviv al, since λ < λ s = 1 / (2 √ d − 1). Therefore, according to Theorem 2.3, there is a p ositiv e probability of surviving in T q starting from s ome x ∈ X without visiting o (clearly x ∈ T q ). This w ould imply that the process restricted to T q surviv es, but this is false. (4) Similarly , we could prov e that λ ( T d ) = 1 /d . In the end, we hav e λ w = 1 /d < 1 /q = λ ( T 5 ) < 1 / (2 √ d − 1) = λ s . As an example, tak e q = 5 and d = 6. Let K ∗ coincide with K except for the entry k ∗ oo := k > 0. Denote by λ ∗ w , λ ∗ ( T 5 ), and λ ∗ s the critical v alues of the BR W ( X , K ∗ ), viewed as functions of k > 0. Using Corollaries 5.7 and 5.8 and argumen ts analogous to those in [11, Example 1], one can show that there exist constants 0 < k 1 < k 2 < k 3 suc h that • for all k ∈ [0 , k 1 ] we hav e λ ∗ w = λ w < λ ∗ ( T d − 1 ) = λ ( T d − 1 ) < λ ∗ s = λ s ; • for all k ∈ ( k 1 , k 2 ) we hav e λ ∗ w = λ w < λ ∗ ( T d − 1 ) = λ ( T d − 1 ) < λ ∗ s < λ s ; • if k = k 2 w e hav e λ ∗ w = λ w < λ ∗ ( T d − 1 ) = λ ( T d − 1 ) = λ ∗ s < λ s ; • for all k ∈ ( k 2 , k 3 ) we hav e λ ∗ w = λ w < λ ∗ ( T d − 1 ) = λ ∗ s < λ ( T d − 1 ) < λ s ; • if k = k 3 w e hav e λ ∗ w = λ w = λ ∗ ( T d − 1 ) = λ ∗ s < λ ( T d − 1 ) < λ s ; • for all k ∈ ( k 3 , + ∞ ) we hav e λ ∗ w = λ ∗ ( T d − 1 ) = λ ∗ s < λ w < λ ( T d − 1 ) < λ s . While it is p ossible to compute k 1 , k 2 , and k 3 explicitly , the required calculations are length y and offer little additional insight. F urther examples can b e constructed; how ever, they fall outside the scope of the present pap er. F or an interesting application to aging BR Ws, we refer the reader to [13], where our results are used to determine critical v alues for the discrete-time coun terpart of a non-Marko vian pro cess. Remark 6.1. In the ab ove example, we studie d a family of GRBR Ws p ar ametrize d by k > 0 . This le ads to the fol lowing gener al observation b ase d on Cor ol lary 5.5. L et { M t } t ∈ I b e a family of GMBR Ws indexe d by t ∈ I ⊆ R such that [ M t ] = [ M r ] for al l t, r ∈ I . Supp ose that the map t 7→ λ M t s is nonincr e asing and c ontinuous. 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Recurrence for branching Mark ov c hains. Ele ctr on. Commun. Pr ob ab. 2008 , 13 , 576–605. [31] Peman tle, R.; Stacey , A.M. The branching random walk and contact pro cess on Galton–W atson and nonhomo- geneous trees. Ann. Pr ob ab. 2001 , 29 , 1563–1590. [32] Stacey , A.M. Branc hing random wa lks on quasi-transitive graphs. Combin. Prob ab. Comput. 2003 , 12 , 345–358. [33] Su, W. Branc hing random w alks and con tact processes on Galton-W atson trees. Ele ctr on. J. Pr ob ab. 2014 , 19 , 12 pp. [34] Zucca, F. Surviv al, extinction and approximation of discrete-time branc hing random w alks. J. Stat. Phys. 2011 , 142 , 726–753. D. Ber t acchi, Dip ar timento di Ma tema tica e Applicazioni, Universit ` a di Milano–Bicocca, via Cozzi 53, 20125 Milano, It al y. Email address : daniela.bertacchi@unimib.it F. Zucca, Dip ar timento di Ma tema tica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, It al y. Email address : fabio.zucca@polimi.it 18