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  • 5/24/2018 Beznea Deaconu Lupascu Branching Fragmentation

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    Branching processes and stochastic fragmentation equation

    Lucian Beznea1, Madalina Deaconu2, and Oana Lupacu3

    Abstract. We investigate branching properties of the solution of a stochastic differential equationof fragmentation (SDEF) and we properly associate a continuous time cdlg Markov process on thespace S of all fragmentation sizes, introduced by J. Bertoin. A binary fragmentation kernel inducesa specific class of integral type branching kernels and taking as base process the solution of the initial(SDEF), we construct a branching process corresponding to a rate of loss of mass greater than a givenstrictly positive size d. It turns out that this branching process takes values in the set of all finiteconfigurations of sizes greater than d. The process on S is then obtained by letting d tend to zero.A key argument for the convergence of the branching processes is given by the Bochner-Kolmogorovtheorem. The construction and the proof of the path regularity of the Markov processes are basedon several newly developed potential theoretical tools, in terms of excessive functions and measures,compact Lyapunov functions, and some appropriate absorbing sets.

    Mathematics Subject Classification (2010): 60J80, 60J45, 60J40, 60J35, 47D07 60K35

    Key words: Fragmentation equation, fragmentation kernel, stochastic differential equationof fragmentation, discrete branching process, branching kernel, branching semigroup, excessivefunction, absorbing set, measure-valued process.

    1 Introduction

    We study branching properties of the solution of a stochastic differential equation of fragmen-tation. Recall that the basic property of a measure-valued branching process is the following:

    if we consider two independent versions XandX

    of the process, started respectively from twomeasures and , thenX+ X and the process started from + are equal in distribution.In studying the time evolution of fragmentation phenomena, it is supposed that "fragmentssplit independently of each other", so, a branching property is fulfilled; cf. [4]. More specific,a main tool for defining the fragmentation chains are the branching Markov chains.

    A different stochastic approach for studying the fragmentation (and coagulation) phenomenawas developed in [21, 22, 27]: the evolution of the size of a typical particle in the system duringa fragmentation process may be described by the solution of a stochastic differential equation,calledstochastic differential equation of fragmentation(SDEF).

    In this paper we associate a continuous time Markov branching process to an (SDEF),describing the time evolution of the fragments greater than a strictly positive size d. The

    1Simion Stoilow Institute of Mathematics of the Romanian Academy, Research unit No. 2, P.O. Box 1-764,

    RO-014700 Bucharest, Romania, and University of Bucharest, Faculty of Mathematics and Computer Science.

    E-mail: [email protected], Villers-ls-Nancy, F-54600, France;

    Universit de Lorraine, CNRS, Institut Elie Cartan de Lorraine - UMR 7502, Vandoeuvre-ls-Nancy, F-54506,

    France. E-mail: [email protected], Villers-ls-Nancy, F-54600, France; Universit de Lorraine, CNRS, Institut Elie Cartan de Lorraine

    - UMR 7502, Vandoeuvre-ls-Nancy, F-54506, France and Simion Stoilow Institute of Mathematics of the

    Romanian Academy. E-mail: [email protected]

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    model for the time evolution of all fragments (of arbitrary small size) is then constructed as alimit of a sequence of branching processes, corresponding to a fixed sequence of sizes (dn)n1decreasing to zero. It is a continuous time Markov process on the state S of all fragmentationsizes, considered by J. Bertoin. This process should be compared with thestochastic coalescentprocess, induced by Smoluchowskis coagulation equation in [36], page 95.

    As a byproduct we emphasize integral type branching kernels on the space of all finite con-

    figurations of an interval[d, 1], associated to the given fragmentation kernel and correspondingto the rate of loss of mass (in sense of [27]) greater than a fixed size d. These branching kernelslead to relevant examples of branching processes and it is possible to write down the nonlinearevolution equations satisfied by the associated cumulant semigroups.

    The paper is organized as follows.In the next section we present the fragmentation equation and the stochastic differential

    equation associated to it, following mainly [27]. A binary fragmentation kernelF is fixed, westate some hypotheses, give the basic definitions, and an example. Corollary 2.2 points out thatin the case when the uniqueness of the solution holds, the solution of the stochastic differentialequation of fragmentation induces a standard (Markov) process with state space the interval[0, 1], its transition function being a C0-semigroup onC([0, 1]).

    In Section 3 we first prove some properties of the real-valued Markov processes, producedby the procedure presented in Section 2, from the fragmentation kernel F truncated to sizesgreater thandn. In particular, the interval [dn, dn1)becomes an absorbing set and therefore itis possible to restrict the process to this set. Putting together all these restrictions we obtain thebase process of a forthcoming branching process on the setEn of all finite configurations of thesetEn:= [dn, 1],n 1. We show in Section 4 (Proposition 4.6) that the associated sequence oftransition functions is a projective system and then, applying the Bochner-Kolmogorov theorem,we obtain a transition function on S (Proposition 4.7). The already mentioned branchingkernels associated to the given fragmentation kernelF, necessary for constructing the branchingprocesses, are also introduced in Section 4.

    The results on the existence of the branching processes (with state spacesEn, n 1) andof the fragmentation process (with state spaceS) are proved in Section 5, Proposition 5.1 andTheorem 5.2. A fragmentation property of the Markov process with state spaceS is provedin Corollary 5.4. We apply the main result from [11] and a method developed in [12], using aRay type compactification technique. A key point in proving the existence of the fragmentationprocess and its path regularity is the fact that there are excessive functions having compactlevel sets (see (5.13) and Remark 5.3).

    The paper is completed by two appendices. Appendix (A) gives briefly some necessary com-plements on the potential theory associated to a right (Markov) processes: the entry time, thereduced function, excessive and strongly supermedian functions, absorbing sets, the restrictionto an absorbing set of a resolvent and of a process, excessive measures. Appendix (B) presents

    the proofs of two results from Section 4.

    2 Fragmentation equation and SDE

    In this section we introduce the stochastic differential equation associated to the fragmentationequation.The stochastic model. We consider a model which describes the fragmentation phenomenonfor an infinite particle system. Each particle is characterized by its size and, at some random

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    times, it can split into two particles by conserving mass. Let us denote by c(t, x) the concen-tration of particles of size x at timet in the system. The evolution in time ofc(t, x)is governedby the fragmentation equation:

    (2.1)

    tc(t, x) =

    1x

    F(x, y x)c(t, y)dy 1

    2c(t, x)

    x0

    F(y, x y)dy

    for all t 0 and x [0, 1],

    c(0, x) =c0(x) for all x [0, 1].

    In equation (2.1), F is the fragmentation kernel, that is F : (0, 1]2 R+ is a symmetricfunction and F(x, y) represents the rate of fragmentation of a particle of size x+y into twoparticles of sizexand y . We can suppose that the size of the initial particle is one.

    In the first line of (2.1), the first term on the right hand side is counting the creation ofparticles of size x, due to the fragmentation of particles of larger size, say y, with y > x, intotwo parts x and y x. The second term counts for the particles of size x which disappearsafter splitting into two smaller particles of size y and x y, fory < x.

    We aim to introduce a pure jump Markov process on R+denoted by(Xt)t0whose law is the

    solution, in some sense, to the equation(2.1). This process will describe the evolution of the sizeof a typical particle in the system. The stochastic approach of the coagulation/fragmentationmodels goes up to the works of Deaconu, Fournier, and Tanr [21, 22] where mainly the co-agulation part was considered. This phenomenon is more complex as it leads to non-linearequations. Later on, Fournier and Giet [27] considered the coagulation/fragmentation modeland obtained existence results for the case of an infinite total rate of fragmentation and alsothey allow existence of particles of mass zero. Other studies on pure fragmentation case weremade by Bertoin [2, 3], allowing multiple fragmentation and also erosion. Haas [28] studied theappearance or not of mass-zero particles.

    We follow here mainly the structure of [27] for the pure fragmentation phenomena that we

    aim to link to the branching processes.The main point that allows a probabilistic approach of (2.1) is given by the conservation

    of mass property. This writes on the form10 xc(t, x)dx= 1 and means that p(t, x) =xc(t, x),

    x [0, 1], is a probability distribution for every fixed t. The aim is to describe the processhaving this distribution.

    We start by stating some hypotheses on the fragmentation kernel.

    Hypothesis

    (H1) The fragmentation kernelF : (0, 1]2 R+ is a continuous symmetric map. Moreover,

    F is continuous from [0, 1]

    2

    toR

    + {+}. Let us define the function

    (x) =

    1

    x

    x0

    y(x y)F(y, x y)dy forx >0,

    0 forx = 0,

    which is supposed continuous on[0, 1]. (x)represents therate of loss of mass of particlesof massx. For each (0, 1) we have

    (2.2) limk

    supx

    k(x) = 0,

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    where

    (2.3) k(x) := 1

    x

    x0

    y(x y)F(y, x y) {F(y,xy)k}dy, k N.

    Notion of solutionAssume that (H1) holds. A family (Qt)t0 of probability measures on [0, 1] is solution of

    (2.1) if the following conditions is fulfilled :

    (2.4) Qt, = Q0, +

    t0

    Qs, Fds, for all C1([0, 1]) and t 0,

    where we denote Qt, =10

    (y)Qt(dy) and for any x [0, 1] we define

    (2.5) F(x) =

    x0

    [(x y) (x)]x y

    x F(y, x y)dy.

    Note that under hypothesis(H1), limx0+

    F(x) = 0.

    In order to construct the pure jump stochastic process associated to (2.4) we introduce aprobability space(, G, (Gt)t0,P). Consider also D([0, +), [0, 1])the space of cdlg functionsfrom[0, +) into [0, 1], endowed with the Skorokhod topology.

    Under the hypothesis (H1), let Q0 be a probability measure on [0, 1]. We say that X isa solution of the stochastic differential equation of fragmentation (abbreviated (SDEF)) if thefollowing conditions hold :

    1. X= (Xt)t0is an adapted process on(, G, (Gt)t0,P)whose paths belong toD([0, +), [0, 1]

    2. L(X0) =Q0.3. For allT0,E

    supt[0,T] |Xt|

    p+1

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    Remark. IfQt, t0, has a density with respect to the Lebesgue measure on [0, 1] and if wesetc(t, x) := dQt

    xdx, thenc(t, x) is a solution of (2.1); see [23].

    We give an example of a fragmentation kernel which satisfies the hypothesis (H1).

    Example. An example of a fragmentation kernelF : [0, 1]2 R+which satisfies the hypoth-

    esis(H1) is F(x, y) =x + y. The rate of loss of mass of particles of mass x is

    (x) = 1

    x

    x0

    y(x y)F(y, x y)dy=x3

    6, x >0.

    Clearly is continuous on [0, 1] and the fragmentation equation (2.1) becomes

    tc(t, x) =

    1x

    yc(t, y)dy x2

    2c(t, x) for all t, x 0,

    c(0, x) =c0(x) for all x 0.

    Observe that the mass conservation condition 10 xc(t, x) = 1is equivalent with c(t, 0) =t + 2,t 0.The uniqueness of the solution for the coagulation/fragmentation equation was studied by

    Banasiak and Lamb in a series of papers [1, 33]. Their approach is based on the semigrouptheory. For the pure fragmentation case, under polynomially bounded fragmentation conditionsthey obtain uniqueness. For the discrete mass case, in the coagulation/fragmentation context,the uniqueness was also studied by Jourdain [31, 32] by using a probabilistic interpretation.

    We emphasize now the Markov process induced by the solution Xof the stochastic differ-ential equation of fragmentation, in the case when the uniqueness of the solution holds.

    For each x [0, 1] let Xx = (Xx,t)t0 be the solution of the stochastic differential equationof fragmentation with the initial distribution x, i.e.,Q0= x.

    Corollary 2.2. Assume that for eachx [0, 1], takingQ0 = x, the equation (2.4) has a uniquesolution(Qt,x)t0 and the function [0, 1] x Qt,x, is continuous for each C

    1([0, 1])andt >0. Then the family of kernels(Qt)t0 on [0, 1], defined as

    Qtf(x) :=Qt,x, f, fpB([0, 1]), x [0, 1],

    induces aC0-semigroup onC([0, 1]) and consequently it is the transition function of a standard(Markov) process X0 = (, F, Ft, X

    0t , P

    x) with state space [0, 1]. In addition, the followingassertions hold.

    (i) For allt 0 andx [0, 1] (Xx,t,P) and(X0t , P

    x) have the same distribution.

    (ii) For everyt >0 we have a.s. X

    0

    t X

    0

    0 .Proof. The semigroup property of (Qt)t0 is rather a straight-forward consequence of theuniqueness. Indeed, we have to show that Qt+t= Qt(Qt), so, it is enough to prove that themappings Qt+s,x verifies the equation (2.4) (with Q0 = x, x [0, 1]) and Qt instead of. We have

    Qt+t,x =(x) +

    t0

    Qs,x, Fds +

    t0

    Qt+s,x, Fds

    =Qt(x) +

    t0

    Qt+s,x, Fds.

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    We claim that(Qt)t0 is aC0-semigroup onC([0, 1]). Indeed, if C1([0, 1])thenF(x)

    ||||(x), x [0, 1], and by (2.4) ||Qt || t|||||||| for all t >0. The assertionfollows by the density ofC1([0, 1])in C([0, 1]). The existence of the standard processX0 having(Qt)t0 as transition function is now a consequence of a main result on Feller processes, seee.g., [17], Theorem (9.4).

    Assertion(i)is clear since by Proposition 2.1, for each x [0, 1], we haveL(Xx,t) =Qt,x=

    L(X0t). (ii) is a consequence of(i), observing that from (2.6) for each x [0, 1] and t >0 we

    get a.s. Xx,t Xx,0.

    Remark. Let(L, D(L)) be the infinitesimal generator of theC0-semigroup (Qt)t0 from Corol-lary 2.2. ThenC1([0, 1]) D(L) and the restriction ofLto C1([0, 1]) is the operatorFgiven by(2.5). In particular, for every C1([0, 1]) and every probability on [0, 1] the process

    (X0t)

    t0

    F(X0s )ds, t 0,

    is an (Ft)t0-martingale under P :=

    Px(dx). (The martingale property is a version of a

    result from [27], page 1313.)Indeed, observe first that if C1([0, 1]) then F C([0, 1]) (as in [27], page 1314).

    Consequently, from theC0-continuity of(Qt)t0 we deduce that the function (s) :=||QsF F||,s R+, is continuous and therefore, using (2.4),

    limt0

    Qt t F

    limt0

    1

    t

    t0

    (s)ds= 0.

    We conclude that D(L) and L = F. The claimed martingale property is a straight-forward consequence of the Markov property of the process X0.

    3 Markov processes induced by fragmentation kernelsLetX = (, F, Ft, Xt, Px) be a right (Markov) process with state space E, a Borel subset of[0, 1], and let (Pt)t0 be its transition function,

    Ptf(x) =Ex(f Xt), x E, fpB(E);

    B(E) denotes the Borel -algebra of E and pB(E) (resp. bpB(E)) the set of all positivenumerical (resp. bounded)B(E)-measurable functions on E. Assume thatX is conservative(i.e.,Pt1 = 1) and for x E and t 0 letPt,x be the probability measure on (E, B) inducedby the kernel Pt,

    Pt,x(A) :=Pt(1A)(x) for all A B(E).

    Let furtherU= (U)>0 be the resolvent family ofX,

    Uf(x) =Ex

    0

    etf Xtdt=

    0

    etPtf(x)dt, fpB(E), >0.

    A set A B(E) is called absorbingprovided that RE\A 1 = 0 on A; see (A1) in Appendix(A). It is easy to see that the property of a set to be absorbing does not depend on and thatevery absorbing set is finely open. Recall that the fine topologyis the smallest topology on E

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    making continuous allU-excessive functions. Note that the absorbing sets we consider are notnecessary finely closed; see e.g. [38] and (3.1) from [15].

    IfA is absorbing andx A thenPx(Xt A) = 1for all t 0, i.e., the probability measurePt,x is carried by A, or equivalently Pt(1E\A) = 0 on A for all t 0.

    We fix a sequence (dn)n1 (0, 1) strictly decreasing to zero and for each n 1 define

    En:= [dn, 1].

    The main hypotheses are the following.

    (H2) For each n 1 there exists a conservative right Markov process Xn with state space Enand transition function (Pnt )t0 such that

    Pn+1t,x =Pnt,x for all n 1, t 0, andx En.

    (H3) For eachn 1 the set

    E

    n:= [dn+1, dn)is absorbing in En+1 with respect to the resolventUn

    +1 = (Un+1 )>0 ofXn+1.

    Remark. (i) The compatibility betweenPn+1t andPnt stated in(H2)expresses the fact that the

    Markov processXn is induced by a fragmentation in particles with "size" bigger thandn.(ii) With the above interpretation, condition(H3) is natural: if a particle is already smaller

    thandn, then it is not possible to produce further "fragments" with bigger size.

    A first consequence of the hypotheses (H2) and (H3) is the following.

    (3.1) The setEn is absorbing in En+1 (with respect to the resolvent Un+1).Indeed, ifx En then by(H2), P

    n+1t,x (En) =P

    nt,x(En) = 1, henceP

    n+1t,x (E

    n) = 0 for allt >0,

    Un+1 (1En) =

    0

    etPn+1t (1En)dt= 0 on En.

    So, the function v := Un+1 (1En) is Un+1 -excessive and vanishes onEn. Since by(H3) the set

    En is absorbing in En+1 it is a finely open subset ofEn+1 and thus Un+1 (1En)> 0 on E

    n and

    we conclude that En= [v= 0], therefore it is absorbing by (A1.4) from Appendix (A).

    LetFbe a fragmentation kernel as in Section 2 and for n 1 define

    Fn(x, y) := 1(dn,1](x y |x y|)F(x, y), x, y E:= [0, 1].

    Assume further that hypothesis (H1) is fulfilled by the fragmentation kernel F. We claimthat

    (3.2) condition(H1) is verified by Fn for all n.

    Indeed, the continuity of the corresponding rate of loss of mass follows by dominate convergencewhile(2.2) is fulfilled because FnF.

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    By (3.2) and Proposition 2.1 for every n 1 and each probability Q0 on [0, 1] there existsolutions to the (SDEF) and to(2.4), where in(2.6) F is replaced byFn. Assume that:

    (3.3) Taking Q0 = x, the equation (2.4) has a unique solution (Qnt,x)t0 and the function

    [0, 1] x Qnt,x, is continuous for each C1(E) and t >0.

    By (3.3) and Corollary 2.2 there exists a standard process X0,n

    with state space E andtransition function (Qnt)t0, where Q

    nt f(x) := Q

    nt,x, f, f pB(E), x E, and (Q

    nt)t0 is a

    C0-semigroup onC(E).Assertion(ii) of Corollary 2.2 implies that for all n 1

    (3.4) X0,nt X0,n0 a.s. for eacht >0.

    LetU0,n = (U0,n )>0 be the resolvent of(Qnt)t0,U

    0,n =

    0 e

    tQntdt, >0.

    Proposition 3.1. The following assertion hold forn 1 and with respect to the resolventU0,n

    onE.

    (i) For allx dn andt 0 we haveQn

    t,x= x.(ii) The setEn= [dn, 1] is an absorbing subset ofE.(iii) For eachx [0, 1) the sets [0, x] and [0, x) are absorbing subsets ofE. In particular,

    the setEn1 = [dn, dn1) is absorbing, whereE0:= E1.

    Proof. If C1(E) let

    Fn(x) :=

    x0

    [(x y) (x)]x y

    x Fn(y, x y)dy, x E.

    Ifx dn then Fn(y, x y) = 0 for all y, hence Fn(x) = 0 and by (2.4) Qnt,x = x, hence (i)holds. If in addition supp [0, dn]thenFn= 0and again by(2.4) Qnt = . Consequently,Qnt(1[0,dn)) = 1[0,dn) for all t0, U0,n (1[0,dn)) = 11[0,dn), En = [U

    0,n (1[0,dn)) = 0], therefore (ii)

    also holds.Assertion (iii) is a consequence of(3.4). Indeed, observe first that the right continuity of

    the trajectories implies that for all x Ewe have Px-a.s.

    X0,nt X0,n0 for allt 0

    and consequently for each x E

    X0,nt x for all t 0 , Px a.s.

    It follows that D(x,1] = ,Px-a.s. and by (A1.2) we conclude that the set [0, x] is absorbing.The set[0, x) is also absorbing by (A1.3), since [0, x) =

    n[0, (x

    1n

    )+].

    Because by assertion (iii) of Proposition 3.1 the set En1 is absorbing with respect to the

    resolvent U0,n, n 1, we may consider the restrictionX0,n of X0,n to En1; see (A1.5) inAppendix (A). It is a conservative standard (Markov) process with state space En1 and let

    ( Pnt )t0 be its transition function. By (A1.6) we havePnt (f|En1) =Qnt f on En1 for each fpB(E) and t 0.8

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    Let now n 1 be fixed. Then En =n

    k=1 Ek1 and we consider the conservative standard

    process Xn with state space En, which behaves asX0,k on Ek1, 1 k n. The transitionfunction(Pnt )t0 ofX

    n is the following

    (3.5) Pnt (f|En) =

    Pkt(f|Ek1) =Q

    kt f on E

    k1, 1 k n, fpB(E), t 0.

    Proposition 3.2. Conditions(H2) and(H3) are fulfilled byXn

    , n 1, and

    Xnt Xn0 a.s. for eacht >0.

    Proof. (H3) holds since the process Xn is by construction such that all the sets Ek,k n 1,

    are absorbing in En. Ifx En then there exists k n such that x Ek1 and by (3.5) for

    fpB(E) and t 0 we have

    Pn+1t,x (f|En+1) =Qkt f(x) =P

    nt,x(f|En)

    and consequently(H2) also holds. The claimed inequality follows from (3.4).

    4 Branching kernels and transition functions on the space

    of fragmentation sizes

    For a Borel subsetEof[0, 1] define the setEof finite positive measures on EasE:=

    kk0

    xk :k N, xkEfor all 1 k k0

    {0},

    where0denotes the zero measure. We identifyEwith the union of all symmetric m-th powersE(m) ofE: E=

    m0

    E(m),

    whereE(0) :={0}; see, e.g., [29, 13, 11]. The setEis called thespace of finite configurations of Eand it is endowed with the topology of disjoint union of topological spaces and the correspondingBorel-algebraB(E); see [25].

    Let M(E) be the set of all positive finite measures on E. For a function f pB(E) weconsider the mappingslf :M(E) R+ and ef :M(E)[0, 1], defined as

    lf() :=, f:= fd, M(E), ef:= exp (lf).Consider the -algebra M(E) on M(E) generated by {lf : f bpB(E)}.E becomes aM(E)-measurable subset ofM(E) and the trace ofM(E) onE isB(E).

    Ifp1, p2 are two finite measures onE, then their convolution p1 p2 is the finite measure onEdefined for every FpB(E) byE

    p1 p2(d)F() :=

    E

    p1(d1)

    E

    p2(d2)F(1+ 2).

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    If pB(E), define themultiplicative function:E R+ as(x) =

    k:xk>0

    (xk), ifx= (xk)k1E,x= 0,1, ifx= 0.

    Observe that each multiplicative function, pB(E), 1, is the restriction toE of anexponential function on M(E), = e ln.In the harmonic analysis on configuration spaces the multiplicative functionis calledcoherentstate; see, e.g., [25].

    Remark 4.1. Since the familyA = {ef : f pbB(E)} is multiplicative, separates the points

    ofM(E), and(A| E) = B(E), the following assertions hold for two finite measuresp1, p2 onE:

    (i) p1= p2 if and only ifp1(

    ) =p2(

    ) for all pB(E), 1.

    (ii) p1 p2() =p1()p2() for all pB(E), 1.Recall that a bounded kernel N on(E, B(E))is calledbranching kernel ifN+=N N for all, E,

    where N denotes the measure on (E, B(E)) such that gdN = Ng() for all g pB(E).Note that ifNis a branching kernel onEthenN0= 0 M(E).(4.1) IfB : pB(E) pB(E) is a sub-Markovian kernel (resp. a Markovian kernel) thenthere exists a unique sub-Markovian (resp. Markovian) branching kernel

    B on(

    E, B(

    E)) such

    that B=B for all pB(E), 1.Sketch of the proof of(4.1). The kernelB is defined as:

    (4.2) Bx:=

    Bx1 . . . Bxn, ifx= x1+ . . . + xn , x1, . . . , xn E,

    0 , ifx= 0.

    Examples of branching kernels.

    1. Let (qm)m1 pB(E) be such thatm1 qm = 1. One can consider the Markovian kernelB: bpB(E)bpB(E) defined as

    Bh(x) :=m1

    qm(x)hm(x , . . . , x), h bpB(E),wherehm:= h|E(m). By (4.1) there exists a branching kernelB onEsuch thatB|E=

    m1

    qmm for each pB(E), 1.

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    2. Branching kernels induced by binary fragmentation kernels. LetF : [0, 1]2 Rbe a symmetric function. Recall that (cf. [27], page 1303) F(x, y) may be seen as the rate offragmentation of particles of mass x+y into particles of mass x, y. F is called fragmentationkernel.

    Letn 1 and define the function 0n: En R+ as

    0n(x) := xdn

    y(x

    y)

    x F(y, x y)dy, x En.

    Observe that0n is continuous on En and limxdn

    0n(x) = 0.

    Analogously with the interpretation given in [27] (where dn= 0),0n(x) represents the rate

    of loss of mass of particles of mass x greater thandn >0. This truncation0n of should be

    compared with n defined in (2.3), which is also a truncation of but it refers rather to thelarge values of the fragmentation kernel F.

    Ifd >0 and g bpB([d, 1]) define the function gd bpB([d, 1]) as

    gd(y) :=g(d)1[0,d)(y) + g(y)1[d,1](y), y [0, 1].

    Define further the kernel Bd: pB([d, 1])pB([d, 1]) as

    Bdh(x) := 6

    x3

    x0

    y(x y)hd(y, y)dy, x [d, 1],

    wherehd(, ) denotes the restriction ofh bpB([d, 1]) to [d, 1](2) ={x1+ x2 :x1, x2 [d, 1]}.

    Clearly, the kernelBd is Markovian (sinceBd1(x) = 6

    x3x0 y(x y)dy= 1) and for each x d

    the probability measure Bd,x is carried by [d, 1](2).

    Letn 1 and consider the kernel Bn :bpB(En)bpB(En) defined by

    Bnh:=n

    k=1

    1Ek1

    Bdkh, h bpB(En),

    whereE0 :=E1 = [d1, 1]. In particular, B1 =Bd1, for each n 1 the kernel B

    n is Markovian,

    and the probability measure Bnx , x En, is carried by E(2)n = {x1 +x2 : x1, x2 En}. The

    main property of the kernelBn is the linear dependence between the image of the fragmentationkernel throughBn and the rate of loss of mass, namely

    BnFx(x) =cn(x)F(dn, 0) + 6

    x20n(x), x E

    n1,

    wherecn(x) := 6x3dn0 y(x y)dy, Fx(x, y) := F(y, x y), x, y En, and Fx is regarded as a

    function onEn, having a non-zero component only on E(2)n .Branching processes. We assume that there exists a branching Markov process with statespaceE, associated with the branching kernelB onE; see [11] for results on the existence ofsuch a branching process. Recall that a right (Markov) process with state spaceE is calledbranching processprovided that its transition function is formed by branching kernels; theprobabilistic interpretation of this analytic branching property was mentioned in the beginning

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    of the Introduction. For further developments see the classical works [29, 30, 39], the lecturenotes [20] and [34], the monograph [35], and the articles [26] and [5].

    Let(Pt)t0 be the transition function of the above branching process andU = (U)>0 itsresolvent of kernels onE.Lemma 4.2. If a subsetA ofEis absorbing with respect to U then

    A is an absorbing subset

    ofEwith respect toU.Proof. By [7] one can see that

    (4.3) A subset of Eis absorbing with respect to Uif and only if it is absorbing with

    respect toU for some >0.Applying Proposition 4.8 from [11], if > 0 there exists > 0 such that for every v

    bE(U0), where U0 is the resolvent on E obtained from Uby a convenient perturbation with a

    kernel (cf. Propostion 4.5 from [11]), the function1 ev isU -excessive. Using now(A1.1)one can see that ifv is strongly supermedian with respect to U0 , then the function 1 ev is

    strongly supermedian with respect toU .Let A be absorbing with respect toU. By (4.3) one can show that A is absorbing with respecttoU0 and letv be a strongly supermedian function with respect to U0 such thatA= [v = 0].

    Consequently,

    vd= 0for all Aand vd >0 if /A. We conclude thatA= [ev = 1]and since 1 ev is strongly supermedian with respect to U , again by (4.3) it follows that the

    setAis absorbing with respect toU.Let(Pnt )t0 be the transition function onEn, n 1, induced by (Pnt )t0 and by the kernel

    Bn associated to a fragmentation kernel F.

    Proposition 4.3. Ifx

    En andt >0 then

    Pn+1t,x =Pnt,x.

    Proof. Ifx= 0 then by(4.2) Pn+1t,0 =0=Pnt,0. Hence we may assume further that x= 0. By

    Remark 4.1 (i) we have to prove that Pn+1t (1En)(x) =Pnt(x)for all pB(En), 1, andxEn.Let hnt be the absolutely monotonic map such thatPnt= hnt(); see [11]. So, we haveto show that hn+1t (1En) =h

    nt()onEn. By Proposition 4.1 from [11] h

    n+1t (1En) =:h

    t is the

    unique solution of the equation

    ht(x) = cPn+1t (1En)(x) + c

    t0

    cPn+1tu(Bn+1hu)(x)du, t 0, x En+1,

    where cPnt

    f := ectPnt

    f with 0 < c < 2. By (H2) we have on En: cPn+1t f =

    cPn+1

    t

    (1Enf) =cPnt(f|En) for all f bpB(En+1). On the other hand the following equality also holds on En:

    Bn+1hu = Bnhu. From the above considerationsht verifies on En the equationht=

    cPnt + c

    t0

    cPntu(Bnhu)du, t 0.

    Sincehnt() is also a solution of this equation, again by Proposition 4.1 from [11] we concludethatht= h

    nt() on En.

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    For n 1 letLn be the infinitesimal generator of the right Markov process Xn with statespace En and transition function (P

    nt )t0, given by hypothesis (H2). Let further h

    nt , t 0, be

    the absolutely monotonic map such thatPnt = hnt(), pB(En), 1; see the proof ofProposition 4.3. Consider also the associated cumulant semigroup(Vnt )t0,

    Vnt f := ln hnt(e

    f), fbpB(En), t 0;

    see Corollary 4.3 from [11]. In particular,Pnt (ef) =eVnt f for all fbpB(En).Remark 4.4. (i) By Remark 4.2 (ii) from [11], the integral equation verified by (hnt)t0 is

    formally equivalent tod

    dthnt = (L

    n c)hnt + cBnhnt , t 0.

    (ii) Let n = 1. Since h1t = eV1t f andB1 = Bd1, one can deduce from (i) that the cumulant

    semigroup (V1t )t0 is formally the solution of the following evolution equation

    d

    dt

    vt(x) =evt(x)L(evt)(x)+

    c

    1

    d21(3x 2d1)

    x3 evt(x)2vt(d1)

    6

    x3

    xd1

    y(x y)evt(x)2vt(y)dy

    ,

    forx d1, t 0, wherevt(x) := V1t f(x). The above equation may be compared with the one

    satisfied by the cumulant semigroup of the discrete branching process in the caseL = andwith the first example of a branching kernel, given by a sequence (qn)n1 bpB(E); see [11],Remark 4.4 (iii). Recall also that in this case the equation of the cumulant semigroup (Vt)t0is: d

    dtVtf= Vtf (Vtf)

    2, t 0, see, e.g. [26] and [5].

    The space of fragmentation sizes. Following [4] we consider as state space the set S

    ofdecreasing numerical sequences bounded above from 1 and with limit0,

    S :={x= (xk)k1[0, 1] : (xk)k1 decreasing, limk

    xk = 0}.

    Recall that a sequence x from S may be considered as "the sizes of the fragments resultingfrom the split of some block with unit size" (cf. [4], page 16).

    Let further

    S :={x= (xk)k1 S : k0 N

    s.t. xk0 >0 and xk = 0 for allk > k0}.

    S

    i :={x= (xk)k1 (0, 1] : (xk)k1 decreasing, limk xk= 0}.We have

    S =S Si {0} and S Si =,

    where0denotes the zero constant sequence.Recall that En = [dn, 1], n 1, where (dn)n1 (0, 1) is a sequence strictly decreasing to

    zero and let

    Sn:=

    kk0

    xk :k0 N ; xkEn, for all1 k k0

    .

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    It is convenient to identify a sequence x = (xk)k1 from S with the -finite measure x on

    [0, 1], defined as

    x :=

    k:xk>0

    xk , ifx=0,

    0 , ifx= 0.

    Consequently, the mapping x x identifies S with n1

    Sn. For x S we write x = x

    where it is necessary to emphasize the identification of the sequence x with the measure x.To each set of finite measures Sn on En we add the zero measure 0= 0,

    S0n:= Sn {0}, n 1.

    So, En= S0n for alln 1.Define the mapping n: S

    S0n as

    n(x) :=x|En , x= x S.

    We have n(0) = 0 and n|S0n =IdS0n. Define

    S:= {(xn)n1

    n1

    S0n: xn =n(x

    m) for allm > n 1}.

    In the next two propositions we identify first S andS. We show then that the (branching)

    transition functions we constructed on eachEn,n 1, induces a projective system of probabilitymeasures; for the proofs see (B1) and (B2) in Appendix (B).

    Proposition 4.5. The mappingi: S S, defined as

    i(x) := (n(x))n1, x S,

    is a bijection.

    Proposition 4.6. Let x S and xn := n(x) En, n 1. If t > 0 then the sequence ofprobability measures(Pt,xn)n1 is projective with respect to (En, n)n1, that is

    Pn+1t,xn+1 1n =

    Pnt,xn for alln 1.

    Proposition 4.7. Assume that conditions(H2) and(H3)hold. Then there exists a Markoviantransition function(Pt)t0 on(S, B(S)) such that for eachx S andn 1 we have

    Pt,x 1n =Pnt,xn,wherexn:= n(x).

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    Proof. We apply the Bochner-Kolmogorov theorem (cf. Theorem 53 from [24] and also [19]),which is a more general version of Kolmogorov theorem on the existence of the limit of a pro-

    jective sequence of measure spaces, assuming no continuity of the mappings; for the topologicalcase (with continuity conditions) see [16], Theorem 2.1 in [18] and [37].

    By Proposition 4.6 the system (

    En,Pnt,xn , n)n1 is projective. We already mentioned that

    eachEn is endowed with the canonical topological structures and the corresponding Borel -algebraB(En). Therefore there exists a unique probability measurePt,x on S such that theclaimed equality holds. Note that by Propostion 4.5 the map i: S S is a bijection andby Proposition 1.1 from [19] it identifies S and S as measurable spaces. The uniqueness

    property implies that the family of kernels (Pt)t0 is a transition function onS.5 Fragmentation and branching processes on finite config-

    urations

    LetXn,n 1, be the Markov processes constructed in Section 3 from the fragmentation kernel

    F. In particular, by (3.2) and Proposition 3.2 conditions (H1), (H2), and (H3) are fulfilled byFn and the processes X

    n. We also consider the branching kernels Bn, n 1, associated to F.

    Proposition 5.1. Letn 1 and(Pnt )t0 be the transition function onEn induced by(Pnt )t0and by the branching kernelBn. Then there exists a branching standard (Markov) process with

    state spaceEn, having(Pnt )t0 as transition function.Proof. Note that by(H3)and (3.1) Ekand E

    kare absorbing subsets ofEnfor all k= 1, . . . , n1.

    Consequently, we have

    (5.1) If v0, v1, . . . , vn1 E(Un ), > 0, then the function v :=n1k=01Ekvk is also U

    n -

    excessive; recall that E0= [d1, 1].Consider the vector space Cn defined as

    Cn:= {f : [dn, 1] R : f|Ek

    C(Ek) s.t. limydk

    f(y) R, k= 0, . . . , n 1}

    and letAn denotes the closure in the supremum norm of the linear space [bE(Un )]spanned bythe bounded Un -excessive functions. An does not depend on >0; see e.g. Remark 2.1 from[11].

    We claim that

    (5.2) Cn An.

    To prove it, we start with a notation: ifk {0, . . . , n 1}and f Cn, we consider the functionfk C(En) defined as

    fk(x) =

    f(x) , ifx Ek,

    f(dk+1) , ifdn x < dk+1,

    limydkf(y), ifdkx 1.

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    TheC0-continuity of the semigroups (Qkt )t0 and(3.5) imply that

    lim

    Unfk=f

    uniformly onEk, 0 k n 1. We haveUnfk[bE(U

    n )]for alland k. By (5.1) the function

    v:=n1k=0

    1Ek

    Unfk

    belongs to [bE(Un )] for all >0. We conclude that lim

    v =funiformly on En =n1

    k=0Ek,

    hencef An.We show now that

    (5.3) Pnt (Cn) Cn, t 0, andlimt0

    ||Pnt f f||= 0 for all f Cn,

    i.e., (Pnt )t0 is a C0-semigroup of contractions on Cn. Indeed, iff Cn then again by the C0-

    continuity of the semigroups (Qnt)t0 and (3.5) we have P

    nt f|Ek =Q

    nt fk|Ek and Q

    nt fk C(En),

    so,limt0

    Pnt f=f uniformly on Ek for all k = 0, . . . , n 1, lim

    t0||Pnt f f||= 0.

    Observe that the integral form of the kernels Bdk , occurring in the definition of the kernelBn, implies that

    (5.4) if pB(En), 1,then Bn Cn.LetKn be the kernel on En defined as

    Knf := c

    c + 2Bn(lf), fbpB(En).

    Because the probability measure Bnx is carried by E(2)n , l1|E(2)n = 2, and c 2, we deduce that

    Kn is a sub-Markovian kernel on En. We may consider the perturbation with the kernelKn ofthe semigroup (Pnt )t0, that is, the sub-Markovian semigroup of kernels (Q

    0,nt )t0 on En such

    that for eachfpbB(En) the function rt:= Q0,nt fis the solution of the integral equation

    rt= cPnt f+ c

    t0

    cPntu(Knru)du, t 0;

    see Proposition 4.5 from [11]. By (5.3) and (5.4) we have

    (5.5) (Q0,n

    t )t0 is aC0-semigroup of contractions on Cn.

    LetU0 = (U0)>0 be the resolvent of kernels on En associated with (Q0,nt )t0.

    We can prove now that the following condition()holds forCn (which is a subset ofAn by(5.2)).

    () There exists a countable subsetF0 ofbE(U0)which is additive, 0 F0, and separates thefinite measures onEn, such that{eu :u F0} Cn, Pnt , P

    nt (B

    n) Cn for all Cn,0 1, andt >0.

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    Indeed, the existence of the countable set F0follows using Ray cones techniques (see Proposition2.2 (ii) from [15]). Note that one can takeF0 Cn and (5.5) ensures that it separates the finitemeasures on En. Since Cn is a Banach algebra, it follows that {e

    u : u F0} Cn. By (5.3)and (5.4) we conclude that condition () is verified. The existence of the claimed standard

    process with state space

    En and having(

    Pnt )t0 as transition function follows now by Theorem

    4.10 from [11].

    We can state now the main result on the existence of an associated Markov process onS. We endow S with the topology induced by the identification with S (endowed with theproduct topology) given by Proposition 1.1 from [19].

    Theorem 5.2. There exists a right (Markov) process with state space S, having cdlg tra-

    jectories, and with(Pt)t0 (given by Proposition 4.7) as transition function.Proof. By Proposition 4.7 we have for all n 1

    (5.6) Pt(f n) =Pnt f n, fbpB(En), t 0.

    Let furtherU = (U)>0 be the resolvent of kernels of the semigroup (Pt)t0. From (5.6) weget

    (5.7) U(f n) =Unf n for all fbpB(En) and >0.As a consequence of (5.7) the following properties holds:

    (5.8) If v E(Un ) then v n E(U).The assertion follows by Hunt approximation Theorem.

    (5.9) If is aU-excessive measure then 1n is aUn -excessive measure. If in addition is a potentialU-excessive measure, = U, then 1n is also a potentialUn -excessivemeasure,= ( 1n ) Un . (For the definition of the excessive measure see (A2) in Appendix(A).)

    We have

    (5.10) B(S) =

    n1

    {f n : fbpB(En)}

    .

    Since clearlyB(En) =(bE(Un )),by (5.8) we get(5.11) B(S) =(bE(U)).

    We claim that

    (5.12) all the points ofS are non-branch points with respect toU, i.e., ifu, v are two

    U-excessive functions then inf(u, v) is also a

    U-excessive function.

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    Indeed, ifu=U(f n), v=U(g n) with f , gbpB(En), then by Proposition 5.1 thefunctionwn= inf(Un f,Un g)isUn -excessive and by (5.7) we haveinf(u, v) = inf(Un fn,Un gn) = wn n. By (5.8) we conclude that inf(u, v) E(U). Using (5.10) and Lemma 1.2.10from [7] it follows (by a monotone class argument) that all the points of S are non-branchpoints.

    Since (5.12) and (5.11) hold, by Theorem 2.1 from [12], to obtain that(Pt)t0is the transitionfunction of a cdlg Markov process with state space S, it remains to show that the followingtwo conditions are fulfilled:

    (5.13) For all x S there exists aU-excessive function vx such that vx(x) < and theset[vx k] is relatively compact for all k 1; such a function vx is called compact Lyapunov

    function.

    (5.14) There exists a countable subsetF of [bE(U)], generating the topology ofS, 1 F,and there exists u0 E(

    U), u0 < , such that if , are two

    U-excessive measures with

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    From (5.14) applied for the resolventUn we get 1n = 1n for all n 1 and by (5.10)we conclude that = . So, (5.14) also holds forUn on S.Remark 5.3. (i) A main argument in proving Theorem 5.2 was the existence of the compact

    Lyapunov functions with respect to the resolvent of kernels

    U on S (see the above condition

    (5.13)). This method was initially used for finding martingale solutions of stochastic partial dif-

    ferential equations on Hilbert spaces (cf. [9]) but it turned out to be efficient in other situationstoo, e.g., to prove existence results for measure-valued branching processes (see [5, 12, 11]).

    (ii)According to Remark 2.2(i)from [12], condition(5.13) is necessary in order to obtain aprocess with cdlg trajectories and it is equivalent with the tightness property of the associatedChoquet capacities; for details see also [10, 14, 15].

    Let(Xt)t0be the Markov process with state space S, having the transition function(Pt)t0(see Theorem 5.2). In the sequel ify = (yk)k1 S

    and x [0, 1], we write y x providedthatyk x for all k 1.

    The next corollary emphasize a fragmentation property of(Xt)t0.Corollary 5.4. For eachx [0, 1], y S, y x, andt 0Xt x Py-a.s.

    Proof. Using Proposition 4.7 we have

    Pyt (Xt x) = lim

    nPy(dnXt x) = lim

    n

    Pt,y(1n ([dn, x])) =limn

    Pnt,yn([dn, x]) = limn

    Pyn(Xnt x) limn

    Pyn(Xnt Xn0) = 1,

    where the above inequality holds because Pyn

    -a.s Xn

    0 =yn x.

    Appendix (A): complements on Markov processes

    (A1) The restriction to an absorbing set. We assume further that U = (U)>0 is theresolvent of a right (Markov) processXwith state space E.

    A function v pB(E) is called U-excessive provided that Uv v for all > 0 andlim

    Uv= v point-wise. Denote by E(U) the set of all U-excessive functions.

    If >0, consider the-level subprocess ofX, recall that its transition function is(et

    Pt)t0and has U := (U+)>0 as associated resolvent. Foru E(U) and every subset A ofE weconsider the function

    RA u:= inf{v E(U) : v u onA},

    called the -order reduced function of u on A. It is known that if A B(E) then RA u isuniversally B(E)-measurable and if moreover A is finely open and u pB(E) then RA u

    pB(E).The reduced function RA u is no longer an U-excessive function, however it is strongly

    supermedian; recall that a positive universally B(E)-measurable function v is called strongly

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    supermedian(with respect to U) provided that

    vd

    vdfor every two finite measures, on Esuch that U U; for details see [7].

    (A1.1) Let v be a positive, B(E)-measurable function (or only a nearly Borel measurablefunction). By [6, 8, 7] the following assertions are equivalent.

    (i) The function v is strongly supermedian with respect to U.

    (ii) RM v v for everyM B(E) (or only for every Ray compact subset M ofE).

    (iii) There exists a family F ofU-excessive functions such that v = infF.(iv) We have v = inf{u E(U) : u v}.

    Recall that ifA B(E) then the entry timeofA is the stopping time DA : [0, ],defined as DA() := inf{t0 : Xt()A}, . The following identification (essentiallydue to G. A. Hunt; see e.g. [24]) of the entry operators and the reduced function on a set holdsfor all A B(E) and u E(U):

    RA u(x) =Ex(eDAu(XDA)).

    (A1.2) Using the above formula on the reduced function, one can check that the followingassertions are equivalent for a set A B(E):

    (i) The set A is absorbing, i.e., RE\A 1 = 0onA.

    (ii) We have Px-a.s. DE\A= for every x A.(iii) There exists a strongly supermedian function v such thatA = [v= 0].

    (A1.3) If(Ak)k is a sequence of absorbing sets then

    kAk is also absorbing.

    (A1.4) Examples. If v is a U-excessive function then the sets [v < ] and [v = 0] are

    absorbing. Indeed, since1 1n

    v on the set [v= ]for everyn 1, it follows thatR[v=] 1 = 0

    on [v 0 ofU to A, i.e., the sub-Markovian resolvent of kernels on(A, B(A)), defined as:

    Uf :=Uf|A for allfpB(A),

    where fpB(E) is such that f|A= f.

    Restriction of the excessive function. A function u pB(A) is U-excessive if and only if thereexists a function u E(U) such thatu= u|A.

    (A1.5) Restriction of the process. If A is absorbing then the restriction of U to A is theresolvent of a conservative right (Markov) process with state space A, called the restriction of

    X to A and we denote it byX: := { : Xt() A for all t 0},Px := Px| for allx A, andXt() := Xt() if (see, e.g., (12.30) in [38]). The main observation is that

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    the transition function ( Pt)t0 ofXis precisely the restriction to A of the transition function(Pt)t0 ofX, that is

    (A1.6) Pt(f|A) =Ptf on A for allfpB(E), t 0.(A2) Excessive measures, the energy functional. LetU = (U)0 be a sub-Markovian

    resolvent of kernels on (E, B(E)), such that the -algebraB(E) is generated byE(U) and allthe points ofEare non-branch points with respect to U, >0.A positive -finite measure onEis called U-excessive, provided that U for all

    >0. Let Exc(U) be the set of all U-excessive measures on Eand recall that if Exc(U)then U as . Let Pot(U) be the set of all potential U-excessive measures: if Exc(U) then Pot(U) if= U, where is a -finite onE.

    Theenergy functional L : Exc(U) E(U) R+ is defined as

    L(, u) := sup{(u) : Pot(U) U }.

    Appendix (B)(B1) Proof of Proposition 4.5. Observe first that: x = 0 if and only ifn(x) = 0 for alln 1. Consequently, ifx S and i(x) = i(0) then x = 0. Let now x,y S,x = 0 = y.Thereforex andy are measures on (0, 1] and ifi(x) =i(y) thenx|En =y|En for all n 1and we conclude thatx= y and x= y.

    Ifx S and1 n < mthenn(x) =x|En =n(m(x))and thus i(x) S.Let now(xn)n1

    n1 S

    0n be such that n(x

    m) = xn for all m > n 1. Ifxn = 0for alln 1 then clearly i(0) = (xn)n1. Suppose that there exists n 1 such that xn = 0. Thenthere exists n0 N such that xn = 0 ifn < n0 and xn = 0 ifn n0. For each n n0 letk(n) N be such thatxnk(n)> 0 and x

    nk = 0 for k > k(n), where x

    n = (xnk)k1 S0n. We have

    (B1.1) xmk =xnk ifm > n andk k(n),

    the sequence (k(n))nn0 N is increasing and let k := supn k(n) N

    {}. Ifk n is such thatk(n)> k(n), then fromn(x

    n) = xn we getxn =xn andthereforexk(n) =x

    n

    k(n) < dn. It follows that limkxk = 0, so, x S and i(x) = (xn)n1. We

    conclude thati is surjective.

    (B2) Proof of Propostion 4.6. We have to show that ifM B(En) then(B2.1) Pn+1t,xn+1(

    1n (M)) = P

    nt,xn(M) for all n 1.

    Observe that 1n (M) =En M. By(H3), (3.1), and from Lemma 4.2 it follows thatEn

    andEn are both absorbing subsets ofEn+1. Therefore(B2.2) Pn+1t,xn(

    En+1 \En) = 0,21

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    and

    (B2.3) Pn+1t,xn (En+1 \En) = 0 ifxnEn.

    If xn = 0 then xn+1

    En and by (4.2)

    Pn+1t,0 = 0 =

    Pnt,0. Therefore, using (B2.3),

    Pn+1t,xn+1(1n (M)) =0(M) = P

    nt,xn(M). So, in the sequel we may assume that xn=0.

    Suppose that n+1(x) = n(x). Consider xn En+1 \En, xn = 0, such that xn+1 =

    xn+ xn. BecausePn+1t is a branching kernel we have

    Pn+1t,xn+1 =Pn+1t,xn

    Pn+1t,xn. Consequently,using(B2.3) and (B2.2),

    Pn+1t,xn+1(1n (M)) =

    Pn+1t,xn+1(En M)

    =

    1 EnM(+ )

    Pn+1t,xn (d)Pn+1t,xn(d)

    = En Pn+1t,xn (d)En Pn+1t,xn(d)1 EnM(+ )=

    En

    1 En()Pn+1t,xn (d)

    En

    1M()Pn+1t,xn(d)

    = Pn+1t,xn(M) =Pnt,xn(M),

    where the last equality holds by Proposition 4.3. From the above considerations we concludethat(B2.1) holds in this case.

    Assume now that n+1(x) = n(x). Then xn+1 = xn En and using (B2.2) and againProposition 4.3 we havePn+1t,xn+1(

    1n (M)) =

    Pn+1t,xn(En M) = Pn+1t,xn(M) = Pnt,xn(M),

    hence(B2.1) is fulfilled.

    Acknowledgements. For the first and the third authors this work was supported by a grant ofthe Romanian National Authority for Scientific Research, CNCS UEFISCDI, project numberPN-II-ID-PCE-2011-3-0045.

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