Strong laws of large numbers for a growth-fragmentation process with bounded cell sizes
aa r X i v : . [ m a t h . P R ] D ec Strong laws of large numbers for agrowth-fragmentation process with boundedcell sizes
Emma Horton ∗ Alexander R. Watson † Growth-fragmentation processes model systems of cells that grow continu-ously over time and then fragment into smaller pieces. Typically, on average,the number of cells in the system exhibits asynchronous exponential growthand, upon compensating for this, the distribution of cell sizes converges toan asymptotic profile. However, the long-term stochastic behaviour of thesystem is more delicate, and its almost sure asymptotics have been so farlargely unexplored. In this article, we study a growth-fragmentation processwhose cell sizes are bounded above, and prove the existence of regimes withdiffering almost sure long-term behaviour.
Keywords.
Growth-fragmentation, law of large numbers, asynchronous exponential growth, celldivision, ergodic theorem, spectral radius, spectral gap, intrinsic martingale, spectrally negativeLévy process, dividend process, skeleton decomposition.
1. Introduction
Growth-fragmentation refers to a collection of mathematical models in which objects– classically, biological cells – slowly gather mass over time, and fragment suddenlyinto multiple, smaller offspring. In recent years, there has been a growing interest inprobabilistic models, in the form of growth-fragmentation processes. These have beendeveloped in the framework of piecewise-deterministic Markov processes by several au-thors [3, 17, 10], and in a very general form by Bertoin [8].In this article, we study a particularly simple growth-fragmentation process wherethe long-term behaviour can be described explicitly. In this process, the cells grow ∗ INRIA, Bordeaux Research Centre, Talence, France. [email protected] † Department of Statistical Science, University College London, UK. [email protected] homogeneous growth-fragmentation process, in which all rates are independent of mass.This homogeneous case was studied in [9], where it was observed that no such simpleasymptotic profile exists. However, Cavalli [15] has shown that a simple change in thedrift, making it piecewise constant, can yield convergence of averages to an explicitasymptotic profile. In this work, we use this approach to take the results a step further:not only do we obtain explicit expressions and show that the system averages exhibitthis convergence, but we are also able to prove this behaviour for the stochastic systemof cell masses via a strong law of large numbers.Work in this field often begins with the growth-fragmentation equation , which in gen-eral form is given by ∂ t u t ( x ) + ∂ x (cid:0) τ ( x ) u t ( x ) (cid:1) + ( B ( x ) + D ( x )) u t ( x ) = Z ( x, ∞ ) k ( y, x ) u t ( y ) d y, with some initial condition u = g . The equation can be usefully rewritten in its weakform, in which we have a collection of measures ( µ t ) t ≥ , started from µ = δ x , satisfying ∂ t h f, µ t i = hA f, µ t i , A f ( x ) = τ ( x ) f ′ ( x )+ Z (0 ,x ) f ( y ) k ( x, d y ) − (cid:0) B ( x )+ D ( x ) (cid:1) f ( x ) , (1)where h f, µ i = R f d µ , τ is the growth rate, B ( x ) = R (0 ,y ) yx k ( x, d y ) is the fragmentationrate, D is the killing rate and k is a kernel describing the masses of offspring cells giventhe mass of the parent.The specific case we will consider is when τ ( x ) = ax { x
2e will shortly see that these models represent an averaged version of the stochasticsystem of cell masses which forms our main object of study. To describe this system, letus start by introducing a stochastic process X started at x , which can be thought of asthe trajectory of a single cell with initial mass x . Write P x for the probability measureassociated with this initial mass. First, let ξ be the process ξ t = y + at + N t X i =1 J i , t < ζ, where y = log x , a > N is a Poisson processwith rate B , and ( J i ) i ≥ are independent, identically distributed jump sizes whose sup-port is contained in [ − log 2 , ζ be an exponential random variable with rate k ,and let ξ be sent to some cemetery state at time ζ . Define Π(d x ) = B P ( J ∈ d x ). Theprocess ξ is a Lévy process with drift a , no Gaussian part, Lévy measure Π and killingrate k . Let ¯ ξ t = sup { ξ s : s ≤ t } , take b = log c and define ξ bt = ξ t − ( ¯ ξ t − b ) ∨ , which is the process ξ reflected above at b .Our single-cell trajectory is given by X t = exp( ξ bt ), the exponential of a reflected Lévyprocess, starting at X = x . This trajectory increases with exponential rate a untilreaching c = e b , has negative jumps and is killed at rate k . We understand that, if ξ t is in the cemetery state, then the same applies to X t . Each jump of size ∆ X t < − ∆ X t and where the trajectory of its mass is given bya copy of X which is conditionally independent given its starting value. By introducingthese new cells at every jump of X , and iterating this process with each successive set ofchildren, we build up the growth-fragmentation process. In terms of Bertoin’s Markoviangrowth-fragmentation processes [8] this is a growth-fragmentation with ‘cell process’ X .In order to describe the masses of each cell, we define Z u ( t ) to be the mass of thecell with label u at time t ; the set of labels u will be described later. The growth-fragmentation process , Z ( t ) = P u δ Z u ( t ) , is a point measure-valued process describingthe masses of all cells alive at each time-point.Crucially, the relation between Z = ( Z ( t )) t ≥ and ( µ t ) t ≥ is given by h f, µ t i = E x [ h f, Z ( t ) i ] , (3)for bounded measurable f , with ρ = Π ◦ g − +Π ◦ ¯ g − , where g ( x ) = e x and ¯ g ( x ) = 1 − e x .In other words, as indicated earlier, µ t represents the average linear behaviour of thegrowth-fragmentation process.There is a vast literature on the long-term behaviour of growth-fragmentation equa-tions and processes (see [9, 37, 21, 6] and [13, 19, 8], respectively, to name just a fewreferences.) From the analytical perspective, the conventional approach is to find λ ∈ R ,a positive function h and a measure ν , such that ν A = λν and A h = λh, (4)3here h ν, h i = 1, using the Krein-Rutman theorem [37, 5, 38] or the Lumer-Phillipstheorem [6]. Generalised relative entropy methods [21, 36] can be used to show thelong-term asynchronous exponential growth h f, µ t i ∼ e λt h f, ν i h ( x ) . (5)On the other hand, from a probabilistic perspective, it is often more convenient to workdirectly with the expectations of the growth-fragmentation process Z to show that theprocess satisfies E x [ h f, Z ( t ) i ] ∼ e λt h ν, f i h ( x ) , (6)which, thanks to (3), agrees with (5). One approach used in [9, 15] is to consider a singletagged particle via a Feynman–Kac formula, and to analyse the Laplace transform of thehitting time of points of this particle. Other recent approaches make use of Harris-styletheorems for nonconservative semigroups [5], and quasi-stationary methods [16].Finally, another approach, [4, 14], which can be seen as a hybrid of the functionalanalytic and stochastic perspectives, involves identifying h through some variation of theanalytic approaches above, using it to derive a stochastic semigroup, and then checkingthe ergodic theorem to obtain a probability measure, which is a simple transformation ofthe eigenmeasure ν , leading to the desired long-term behaviour. This idea is the startingpoint for our work.The main aim of this article is to describe the long-term stochastic behaviour of thegrowth-fragmentation process. To this end, we will prove two theorems that can be seenas stochastic analogues of the deterministic asymptotic (6), and which, indeed, imply it.The first of these theorems is the following strong law of large numbers . Theorem 1.
Assume that
B > k , a > Z ( − log v ) ρ (d v ) and Z v − r ρ (d v ) < ∞ for some r > . Define λ = B − k . Then, there exists a probability measure ν and a random variable M ∞ such that for all x ∈ (0 , ∞ ) and continuous, bounded f , e − λt h f, Z ( t ) i → h f, ν i M ∞ , P x -almost surely and in L ( P x ) . In fact, M ∞ is the limit of the martingale M t = e − ( B − k ) t h , Z ( t ) i , and it has expecta-tion constant in the starting mass of the process: E x [ M ∞ ] = 1. The L ( P x ) convergenceimplies in particular (6) with h = 1, but in fact, we derive (4) and (6), including a ver-sion of (6) with exponential rate, under more general assumptions, as the forthcomingPropositions 12 and 13. These are fundamental to our method of proving Theorem 1.An equivalent version of this theorem has been shown to hold for a wide varietyof branching processes, including branching diffusions on bounded or unbounded do-mains [26], superprocesses [25] and, more recently, a general class of branching Markovprocesses with non-local branching [30]. For a different class of growth-fragmentation4rocesses, a weaker form of this theorem was proven in [11], where convergence in L ( P x )was shown. In this article, we follow the approaches used in [30] and [26] to prove thestronger almost sure convergence.The second theorem provides a similar result in the transient case. To aid comparisonwith the literature, it will be useful to introduce a function κ , which in the context ofhomogeneous growth-fragmentations is called the ‘cumulant’ and is defined as: κ ( q ) = aq + Z (0 , x q ρ (d x ) − ( B + k ) . The function κ is convex and smooth. We define q to be the unique solution of theequation κ ′ ( q ) = 0, or equivalently, q = arg inf κ . Theorem 2.
Assume that
B > k , a < Z (0 , ( − log v ) ρ (d v ) , which implies that q > . Define λ = κ ( q ) = inf κ , and assume further that λ > . Let f : (0 , c ] → R be continuous with f ( x ) = O ( x q ) as x → . Then, e − λt h f, Z ( t ) i → , P x -almost surely and in L ( P x ) . This result can be interestingly compared with [19, Theorem 2.3], which considersa homogeneous growth-fragmentation Z ◦ and takes f ( x ) = f q ( x ) = x q . Under theassumptions of the above theorem, part of Dadoun’s result is that e − κ ( q ) t h f q , Z ◦ ( t ) i → q ≥ ¯ q , where ¯ q > q is the unique solution of ¯ qκ (¯ q ) = κ ′ (¯ q ).The fact that ¯ q > q says, in a quantitative way, that the cell masses in the growth-fragmentation process with reflection decay to zero faster than without reflection. Withthis in mind, it would be interesting to study the speed of cell masses in more detail, asdone by Dadoun for homogeneous processes in [19, Corollary 2.4], and we leave this asa future research topic.We would also like to note several other possible directions for future work. Here,we have taken the killing rate to be constant, but it may be possible to relax thisrestriction while remaining in a tractable class of processes. In particular, it wouldbe interesting to consider killing cells, either instantaneously or at exponential rate,when their mass falls below a certain threshold, to mimic mechanisms for cell deathin biological systems. An alternative direction would be to take a more general Lévyprocess as the underlying trajectory of the cell masses; for instance, it should be possibleto incorporate Gaussian fluctuations into this model with little impact on our results.Finally, as mentioned above, we have assumed here that cell divisions are binary, butthis is far from necessary. Indeed, provided that the number of cells produced at divisionis not too large (an ‘ L log L ’ condition will suffice), arbitrary division and repartition ofmass can be incorporated into this model; one mathematical approach to this is describedin [11, p. 8]. This could prove fruitful in biological models for tumour growth, since it5s known that cancer cells can split into more than two daughter cells. It would alsoallow one to study a phenomenon known as aneuploidy, where a cell contains the wrongnumber of chromosomes due to non-binary fission (see [40] and references therein).The rest of this paper is organised as follows. In section 2 we will analyse the ex-pectation semigroup associated with the process Z , as well as the strongly continuoussemigroup associated with the operator A . We will show that these two semigroups areequivalent, in an appropriate sense, for a certain class of functions. In section 3 we willformally define the growth-fragmentation process and consider the basic behaviour ofthe number of cells in the system. In section 4, we will make use of a spine method todistinguish the regimes of long-term behaviour. Finally, sections 5 and 6 are devoted tothe proofs of Theorems 1 and 2, respectively.
2. The growth-fragmentation semigroup
One way to characterise the growth-fragmentation process is via its expectation semig-roup. By considering the average behaviour of particles under the action of suitable testfunctions, we characterise as the unique solution to an evolution equation.On the other hand, from an analytical perspective, it is common to use the generatordefined in (2) to define a strongly continuous semigroup, usually written ( e t A ) t ≥ , whichuniquely solves the growth-fragmentation equation (1).In this section, we will explore both perspectives and, in order to consolidate them,we will show that the two are equivalent for a certain class of test functions. Hence,roughly speaking, both the analytical and probabilistic semigroups characterise the av-erage behaviour of such growth-fragmentation systems, and can therefore both be usedto analyse their long-term behaviour.We start by considering the expectation semigroup associated with the growth-fragmentationprocess. First recall the underlying reflected spectrally negative Lévy process, ξ b , ob-tained by taking the process with Laplace exponent ψ ξ ( q ) = − k + aq − Z −∞ (1 − e qx ) Π(d x ) , satisfying E [ e qξ t ] = e tψ ξ ( q ) , and reflecting it from above at the level b .Further, recall the growth-fragmentation kernel ρ defined on (0 ,
1) by ρ = Π ◦ g − +Π ◦ ¯ g − , where g ( x ) = e x and ¯ g ( x ) = 1 − e x .As before, we write Z = (cid:0) Z ( t ) , t ≥ (cid:1) for the Markovian growth-fragmentation associ-ated with e ξ b , and let c = e b >
0, which is the reflection level for cell sizes in Z .Define Ψ t f ( x ) = E x (cid:2) h f, Z ( t ) i (cid:3) . In our first proposition, we derive an evolution equa-tion for which Ψ t is the unique solution. We regard Ψ as a semigroup on L ∞ ((0 , c ]), thespace of bounded measurable functions on (0 , c ].6 roposition 3. The semigroup Ψ satisfies the evolution equation Ψ t f ( x ) = f ( xe a · t ∧ T c ( x ) ) − k Z t d s Ψ t − s f ( xe a · s ∧ T c ( x ) )+ Z t d s Z ρ (d v ) (cid:2) Ψ t − s f ( xe a · s ∧ T c ( x ) v ) − v Ψ t − s f ( xe a · s ∧ T c ( x ) ) (cid:3) , (7) where T c ( x ) = a log (cid:0) cx (cid:1) is the first time the deterministic path t xe at reaches c .Moreover, this evolution equation uniquely determines Ψ as a semigroup on L ∞ ((0 , c ]) .Proof. Let τ denote the first time that the initial particle either dies or fragments. Byconsidering the two cases t < τ and t ≥ τ we haveΨ t f ( x ) = f ( xe a · t ∧ T c ( x ) ) e − ( B + k ) t + ( B + k ) Z t d s BB + k e − ( B + k ) s Z −∞ Π(d u ) B × n Ψ t − s f ( xe a · s ∧ T c ( x ) e u ) + Ψ t − s f ( xe a · s ∧ T c ( x ) (1 − e u )) o = f ( xe a · t ∧ T c ( x ) ) e − ( B + k ) t + Z t d se − ( B + k ) s Z −∞ Π(d u ) n Ψ t − s f ( xe a · s ∧ T c ( x ) e u ) + Ψ t − s f ( xe a · s ∧ T c ( x ) (1 − e u )) o . Making a change of variables v = e u yieldsΨ t f ( x ) = f ( xe a · t ∧ T c ( x ) ) e − ( B + k ) t + Z t d se − ( B + k ) s Z Π ◦ g − (d v ) n Ψ t − s f ( xe a · s ∧ T c ( x ) v ) + Ψ t − s f ( xe a · s ∧ T c ( x ) (1 − v )) o . Making another change of variables, v − v , for the second integrand in the final lineabove, and recalling that ρ = Π ◦ g − + Π ◦ ¯ g − , we haveΨ t f ( x ) = f ( xe a · t ∧ T c ( x ) ) e − ( B + k ) t + Z t d s e − ( B + k ) s Z ρ (d v ) Ψ t − s f ( xe a · s ∧ T c ( x ) v ) . Applying Dynkin’s identity [24, Lemma 1.2, Chapter 4, Part I] and using the fact thatthat R v ρ (d v ) = B shows that Ψ t f ( x ) does indeed solve (7).For uniqueness, suppose Ψ t and Ψ t are both solutions to (7) and let ˜Ψ denote theirdifference. Then, since B is finite, it is straightforward to show that there exists aconstant C > | ˜Ψ t f ( x ) | ≤ C Z t sup x ∈ (0 ,c ] | ˜Ψ t − s f ( x ) | d s. Gronwall’s inequality then yields the result.7ith this result we have uniquely characterised the average (linear) behaviour of Z using the evolution equation (7). Part of our goal in this work is to describe ourprobabilistic results (Theorem 1 and (6)) about the growth-fragmentation semigroup interms of the spectral theory of the operator A , defined in (2). However, if we wereto define A as the generator of a semigroup with the same action as Ψ (on the spaceof continuous functions vanishing at infinity, say), then its domain would not be largeenough to contain the eigenfunction we identify. To remedy this, we now show that A (or a suitable extension) can be regarded as the generator of a strongly continuoussemigroup on L ((0 , c ]), which solves the growth-fragmentation equation (1) and agreeswith the semigroup Ψ above on their common domain of definition.In general, questions of existence and uniqueness of a semigroup with generator A arenot straightforward; we refer to [20, 35, 34] for some examples of classical and recentapproaches. Our situation is fairly simple since the fragmentation rate is bounded, butnonetheless it does not seem to be easy to derive from existing results, so we developthe proof below.We first focus our attention on the growth (i.e., transport) part of the operator. Wedefine G f ( x ) = axf ′ ( x ) − ( B + k ) f ( x ) , f ∈ D , where D is the subspace of L ((0 , c ]) consisting of functions f : (0 , c ] → R such that(i) f is smooth with compact support, and(ii) lim x → c f ′ ( x ) = 0.Then, we have the following proposition. Proposition 4.
The operator G : D → L is closable and its closure generates a stronglycontinuous semigroup on L ((0 , c ]) .Proof. The majority of the proof follows the method in [41] so we will only give anoutline of the proof here. We start by considering the operators U t f ( x ) = e − ( B + k ) t f ( xe at ) , f ∈ L ((0 , c ]) , t ≥ , which we define as e − ( B + k ) t f ( c ) whenever t > xe at ≥ c .First note that for each t ≥ U t is bounded. Also, for any continuous function f : (0 , c ] → R with compact support, we have k U t f − f k → , as t → . Since the space of compactly supported continuous functions is dense in L , these obser-vations imply ( U t ) t ≥ is strongly continuous. Hence, its infinitesimal generator is closedand densely defined.Now let f ∈ L ((0 , c ]) be such that t f ( xe at ) is smooth with compact support, andfor h >
0, set ϕ h ( x ) := Z h U s f ( x )d s. ϕ h lies in the domain of the infinitesimal generator of ( U t ) t ≥ . Denoteby G the restriction of this generator to functions ϕ h constructed as above. Then, again,by [23, p.620] G ϕ h = U h f − f, and the infinitesimal generator of ( U t ) t ≥ is ¯ G .Our next step is to show G is closable and its closure equals ¯ G . To show this, we firstshow that ϕ h lies in D . First note that from the definition of ϕ h , it is also smooth withcompact support. Next note that ϕ h ( xe at ) = Z t + ht e − ( B + k )( u − t ) f ( xe au )d u. (8)Hence, ∂∂t ϕ h ( xe at ) = e − ( B + k ) h f ( xe a ( t + h ) ) + ( B + k ) Z t + h e − ( B + k )( u − t ) f ( xe at )d u − f ( xe at ) − ( B + k ) Z t e − ( B + k )( u − t ) f ( xe at )d u. Letting t →
0, we havelim t → ∂∂t ϕ h ( xe at ) = U h f ( x ) − f ( x ) + ( B + k ) ϕ h ( x ) , which, in turn, implies that axϕ ′ h ( x ) − ( B + k ) ϕ h ( x ) = U h f ( x ) − f ( x ) . (9)Now, from the definitions of ϕ h and U h , it is not too difficult to show thatlim x → c ϕ h ( x ) = f ( c ) B + k (1 − e − ( B + k ) h )and lim x → c U h f ( x ) = e − ( B + k ) h f ( c ). Taking the limit as x → c in (9) and using thesetwo equalities yields lim x → c ϕ ′ h ( x ) = 0. Hence ϕ h ∈ D .The above analysis also shows that G ϕ h ( x ) = axϕ ′ h ( x ) − ( B + k ) ϕ h ( x ) = G ϕ h ( x ) . Since ϕ h ∈ D , it follows that G ⊂ G .Finally, for any ϕ ∈ D we have that U h ϕ − ϕ = Z h U s G ϕ d s. Due to [31, Theorem 10.5.2], G is a restriction of ¯ G . Therefore, ¯ G = ¯ G and so G isclosable and generates the semigroup ( U t ) t ≥ .9et F : D → L be defined by F f ( x ) = Z f ( xy ) ρ (d y ) . Then, we can extend the result of the previous proposition to G + F . Corollary 5.
The conclusion of Proposition 4 holds when we replace G by G + F .Moreover, if we define A to be the closure of G + F , then its domain D ( A ) is equalto the domain of the closure of G .Proof. First note that since F it is bounded, it can be extended to an operator withdomain L ((0 , c ]). Further, recall ( U t ) t ≥ from the proof of Proposition 4. We will useTheorem 1 of [42] to prove this corollary, which in our context, is as follows.Suppose the following two conditions are satisfied:(I) U t D ⊂ D and for all f ∈ D , the function t
7→ F U t f is continuous.(II) There exist constants α > γ ∈ [0 ,
1) such that for all f ∈ D Z α kF U t f k d t ≤ γ k f k . Then there exists a unique strongly continuous semigroup ( W t ) t ≥ on L ((0 , c ]) suchthat W t f = U t f + Z t W t − s F U s f d s f ∈ D , t ≥ . Moreover, the generator of ( W t ) t ≥ is the closure of ( G + F ) (cid:12)(cid:12) D and its domain is equalto the domain of the closure of G .Now, (I) follows easily from the definitions of U t , D and F . For (II), note that forany α > Z α kF U t f k d t ≤ B k f k Z α e − ( B + k ) t d t = 2 B k f k B + k (1 − e − α ( B + k ) ) . Choosing α > γ := BB + k (1 − e − α ( B + k ) ) ∈ [0 ,
1) yields the result. Hence, theoperator A with domain D ( A ) generates a strongly continuous semigroup, ( W t ) t ≥ on L ((0 , c ]). Moreover, its domain, D ( A ), is equal to that of the domain of the closure of G .Therefore, the operator A is the generator of a strongly continuous semigroup W =( W t ) t ≥ on L ((0 , c ]) and has D as a core. Fixing g ∈ L ((0 , c ]), if we let µ t be defined,though the Riesz representation theorem, by h W t f, g i = h f, µ t i (where h· , ·i representsthe L inner product), then ( µ t ) t ≥ solves (1) with µ (d y ) = g ( y )d y .In fact, in a certain sense the semigroups W and Ψ are equivalent: Proposition 6.
For each f ∈ L ∞ ((0 , c ]) , k W t f − Ψ t f k = 0 . roof. The proof is identical to that of [18, Theorem 8.1].This provides a version of (3) in the introduction, albeit restricted to bounded func-tions f and initial conditions having L density. This will let us consolidate the behaviourshown in (4) and (6) later on.
3. The growth-fragmentation process
We now turn our attention to the growth-fragmentation process, Z . Although we havealready given a construction of Z in the introduction, in this section we provide anotherconstruction via the Ulam-Harris tree, as it will be convenient in later sections to referto the particles via their labels.To this end, we denote by U = S n ≥ { , } n the Ulam-Harris tree of finite sequences of0s and 1s. For u ∈ { , } n , we interpret the length | u | = n as its generation. Further, for v ∈ { , } , we write uv ∈ { , } n +1 to denote the concatenation of u and v and interpretit as the v th daughter of u . The initial individual will be written as ∅ .Now fix x ∈ (0 , c ] and let P x denote the law of the growth-fragmentation processstarted from an initial particle at x . Now define the random variable ζ ∅ by P x ( ζ ∅ > t ) = e − ( B + k ) t , t ≥ , which we see as the minimum of the fragmentation time and death time of the initialparticle. Setting b ∅ = 0 and d ∅ = ζ ∅ , it follows that the trajectory of the initial particleis given by Z ∅ ( t ) = xe a · t ∧ T c ( x ) , t ∈ [ b ∅ , d ∅ ) , where we recall that T c ( x ) = a log( c/x ). With probability k B + k , ζ ∅ denotes a killing timeof ∅ , in which case the process stops. If not, then ζ ∅ denotes the first fragmentationtime, and then we set b u = d ∅ for each daughter u ∈ { , } of the initial particle andrandomly choose v ∈ (0 ,
1) according to the probability measure ρ (d v )2 B . The masses of thedaughters are then given by xv and x (1 − v ). The lifetime of each daughter is denoted by ζ u , which has the same distribution as ζ ∅ under P xv or P x (1 − v ) . We also set d u = b u + ζ u and the trajectory of the u th daughter is given by Z u ( t ) = ye a · t ∧ T c ( y ) , t ∈ [ b u , d u ) , for y = xv, x (1 − v ). The process then continues iteratively, with each cell u and theinitial mass of its daughters u u Z u ( t )) t ∈ [ b u ,d u ) for each u ∈ U . Setting U t to be the set of individualsalive at time t , i.e. U t = { u ∈ U : t ∈ [ b u , d u ) } , the growth-fragmentation process at time t is given by the collection of atomic measures Z ( t ) = X u ∈U t δ Z u ( t ) . In the introduction and in the previous section, this process was constructed in adifferent way, using the ‘cell process’ ξ and adding new cells at each jump. These two11pproaches are consistent: if ( u , u , . . . ) is a sequence of cell labels with the propertythat, for each n ≥ Z u n +1 ( b u n +1 ) /Z u n ( d u n − ) ≥ /
2, then ξ t = Z u n ( t ) , where b u n ≤ t < d u n , t ≥ , is the cell process.Let us now briefly turn our attention to the growth of the total number of cells in Z . Set λ ∗ = B − k and define N t = h , Z ( t ) i , the number of cells in the growth-fragmentation at time t . Since the branching and killing rates are spatially independent, N is a discrete-space, continuous-time branching process.Recall that a branching process is subcritical, critical or supercritical according towhether E [ N t ] is negative, zero or positive, and that in the subcritical and critical cases,the process becomes extinct (eventually N t = 0) with probability 1. We have the fol-lowing simple consequence of branching process theory: Proposition 7. N is subcritical, critical or supercritical according to whether λ ∗ isnegative, zero or positive. In the supercritical case, the extinction probability is k B .Proof. This follows using classical branching process results [2, §I.5]. The probabilitygenerating function of the offspring distribution is g ( z ) = k + Bz B + k , and equation g ( z ) = z has roots 1 and k B , which implies the extinction probability in the result.We end this section with a short discussion of the relationship between the variousparameters of the growth-fragmentation process. For the particular model we considerin this paper, all of our results can be expressed explicitly in terms of the parameters a, k and ρ , due to the fact that k ρ k = 2 k Π k = 2 B , where k·k indicates the total mass ofa measure. We emphasise that the factor of two appears here due to the fact that twoparticles are produced at every fragmentation event.In a more general model, where we allow particles to fragment into more than twopieces, the total mass of ρ is given by the fragmentation rate multiplied by the averagenumber of offspring, and λ ∗ is replaced by k ρ k − B − k . Moreover, the results of thispaper will still hold but with a slight adjustment of the conditions; the condition B > k in Theorem 1 would become k ρ k − B − k >
0, and an ‘ L log L ’ condition on the numberof offspring would be required, which is redundant in the binary case. One key differencein this more general model is that ρ (or equivalently, the linear evolution equation) nolonger uniquely determines the growth-fragmentation process. For this, one needs towork with a non-linear version of the growth-fragmentation equation, as in section 5.
4. Asymptotic regimes
In the previous section, we determined conditions for the total number of cells to eitherreach zero with probability 1 (the (sub)critical cases) or to survive with positive prob-ability (the supercritical case). We now consider the long-term behaviour of cell masses,with the goal of demonstrating that these settle into an equilibrium in the long-term;12his will be most important in the supercritical case, where it will be the foundation ofTheorem 1, but the results of this section apply regardless of (sub- or super-)criticality.Our main tool will be the analysis of a single tagged cell , whose behaviour will tellus something about the average behaviour of the process Z . We can think of this asfollowing a distinguished line of descent in the cell line, where at each cell division eventwe uniformly pick a daughter cell to follow.The motion of the cell mass of this tagged cell will be determined by an exponentialreflected Lévy process, much like the ‘cell process’ ξ in the introduction; however, theprocess itself is different, since it should represent not the locally largest cell, but atypical cell.Introduce a Lévy process η with drift a and Lévy measure Π η = ρ ◦ ( g − ) − , where g − ( x ) = log x . Equivalently, this Lévy measure can be expressed in terms of that of ξ by Π η = Π + Π ◦ s − , where s ( x ) = log(1 − e x ). (Hence, if Π has density π , then Π η hasdensity π η ( x ) = π ( x ) + π (log(1 − e x )) e x (1 − e x ) − .) Denote by ψ η its Laplace exponent,which can be expressed as ψ η ( q ) = aq + Z ( −∞ , ( e qx −
1) Π η (d x ) = aq + Z (0 , ( v q − ρ (d v ) . Again, as with ξ , we define η b to be this process reflected at b , i.e., η bt = η t − (¯ η t − b ) ∨ Y = exp( η b ). Our goal for the remainder of thissection is the investigate the long-term behaviour of Y , identify the growth-fragmentationsemigroup in terms of the semigroup of Y , and hence prove the asynchronous exponentialgrowth (6). This will be a key building block in the proof of the strong law, Theorem 1.In past work on growth-fragmentation equations [10, 15], a key element of the analysiswas an auxiliary function defined in terms of return times of Y . In our context, we maydefine this as follows. Let S ( y ) = inf { t > Y t = y } , and let H ( y ) = inf { t > S ( y ) : Y t = y } be the first return time to y . Define L x,y ( q ) = E x [ e − ( q − λ ∗ ) H ( y ) ; H ( y ) < ∞ ] . Let q ∗ = inf { q ∈ R : L c,c ( q ) < ∞} and λ = inf { q ∈ R : L c,c ( q ) < } . The quantity λ is the conjectural exponential rate in (6), and in previous works, the behaviour of L was used in order to deduce asymptotic results for the growth-fragmentation semigroup.Our situation is actually a little simpler than [10], and we can avoid this step, but weprovide results on L for the sake of comparison.Our first result pertains to the transience and recurrence of Y . Roughly speaking, theclassification depends on the mean of the underlying Lévy process, η . We distinguish13our possible cases: a < Z ( − log v ) ρ (d v ) , (T) a = Z ( − log v ) ρ (d v ) , (NR) a > Z ( − log v ) ρ (d v ) , (PR) a > Z ( − log v ) ρ (d v ) and Z v − r ρ (d v ) < ∞ for some r > . (ER) Proposition 8.
For t ≥ and x ∈ (0 , c ] , the many-to-one formula Ψ t f ( x ) = E x [ f ( Y t ) e λ ∗ t ] holds. Further, defining q ∗ = λ ∗ + max { inf ψ η , −k Π η k} , we have the following cases.(i) If (T) holds, then Y is transient, λ = q ∗ = λ ∗ + inf ψ η and L c,c ( λ ) < .(ii) If (NR) holds, then Y is null recurrent, λ = λ ∗ , L c,c ( λ ) = 1 and L ′ c,c ( λ ) = −∞ .(iii) If (PR) holds, then Y is positive recurrent, λ = λ ∗ , L c,c ( λ ) = 1 and L ′ c,c ( λ ) > −∞ .(iv) If (ER) holds, then for any < w < r , there exist k > and C > such thatfor all x ∈ (0 , c ] , t ≥ , and f such that f w ( x ) := ( x/c ) w f ( x ) is continuous andbounded, (cid:12)(cid:12) E x [ f ( Y t )] − h f, ν i (cid:12)(cid:12) ≤ k f w k (cid:0) ( c/x ) w + C (cid:1) e − kt , where ν is the invariant distribution of Y . Indeed, we can take k = − ψ η ( − w ) and C = R ( c/x ) w ν (d x ) = − ψ ′ η (0) wψ η ( − w ) .Proof. The proof of the many-to-one formula, Ψ t f ( x ) = E x [ f ( Y t ) e λ ∗ t ], follows in a similarway to the proof of Proposition 3, and we omit the full detail. Indeed, splitting on thefirst time η is either killed or jumps, applying Dynkin’s identity and some simple algebraicmanipulations shows that E x [ f ( Y t ) e λ ∗ t ] also solves (7). Since solutions of (7) are unique,it is equal to Ψ t f ( x ) for each x ∈ (0 , c ] and each t ≥ Y in the differentcases. We will show that Y is transient when (T) holds and that Y is recurrent wheneither (NR) or (PR) hold, and defer the proof of the distinction between positive andnull recurrence to Proposition 10.Differentiating ψ η and letting q →
0, we have ψ ′ η (0) = a + Z (0 , log v ρ (d v ) (10)14he recurrence or transience of Y is equivalent to the recurrence or transience of thereflected Lévy process η b . Considering its paths, we observe that η b is recurrent if andonly if lim sup t →∞ η t = ∞ a.s.; and by [33, Theorem 7.1], this occurs if and only if ψ ′ η (0) ≥
0. Combining this with (10), we see that η b , and hence Y , is recurrent if andonly if (PR) or (NR) holds, and transient if (T) holds.We now consider the values of λ and L c,c ( λ ) for each of the first three cases. Byconditioning on the first time Y jumps away from c , it is straightforward to show that L c,c ( q ) = − a Φ( q − λ ∗ ) k Π η k + q − λ ∗ , if q > q ∗ ∞ if q ≤ q ∗ , (11)where Φ is the right inverse of ψ η . Since q ∗ ≥ −k Π η k , it is clear that the denominatoris positive whenever q > q ∗ . Hence, L c,c ( q ) = 1 for some q ≥ q ∗ if and only if there is asolution to the equation Φ( q − λ ∗ ) = 0 with q ≥ q ∗ .If ψ ′ η (0) ≥
0, then Φ(0) = 0 and so q = λ ∗ gives us the solution we seek. It also followsthat L c,c ( λ ) = P c ( H ( c ) < ∞ ) = 1 since ψ ′ (0) ≥ Y is recurrent.If ψ ′ η (0) <
0, then Φ( q − λ ∗ ) is strictly positive for all q ≥ q ∗ , and so λ = q ∗ and L c,c ( λ ) <
1. To show that q ∗ = λ ∗ + inf ψ η , we argue as follows: ψ η ( q ) + k Π η k = aq + Z −∞ e qx Π η (d x ) ≥ , and the two summands on the right are positive and do not approach zero simultaneously(if q ≥ ψ η = inf q ≥ ψ η ( q ) > −k Π η k .To prove the statements about the derivative of L , note that L ′ c,c ( q ) = − a Φ ′ ( q − λ ∗ )( q − λ ∗ + k Π η k ) − a Φ( q − λ ∗ )( q − λ ∗ + k Π η k ) , and observe that Φ ′ ( q − λ ∗ ) = ∞ if and only if q = λ ∗ + inf ψ η . Since inf ψ η = 0 if andonly if (NR) holds, we obtain that L ′ c,c ( λ ) = −∞ if and only if we are in case (NR),which was the claim.Part (iv) follows by Theorem 2 of Goffard and Sarantsev [28].This completes the proof. Remark . (i) As remarked in the proof, the cases (T), (NR) and (PR) are equivalentto E [ η ] being negative, zero and positive respectively, and the moment conditionin (ER) is equivalent to E [ e rη ] < ∞ .(ii) In previous works, the value of L c,c ( λ ) was used to determine the transience orrecurrence of Y . We do not need this for this result, but we continue the spirit ofthese works by making heavy use of the supermartingale function ℓ ( x ) = L x,c ( λ )in our analysis of the case (T) in section 6.15 .1. Recurrent regimes We now consider the cases (NR), (PR) and (ER) in more detail. Our goal is to find aninvariant measure for the process Y , which will turn out to be an eigenmeasure of thegrowth-fragmentation operator.The methods in the section come from the theory of spectrally negative Lévy processes,of which η is an example, and we begin by defining the scale function W : [0 , ∞ ) → [0 , ∞ )with Laplace transform Z ∞ e − βx W ( x ) d x = 1 ψ η ( β ) , valid for β > Φ(0), where Φ( q ) = sup { p : ψ η ( p ) ≥ q } . We will also use W to denote the(Stieltjes) measure associated with the scale function.For details of the existence and properties of scale functions, we refer to [33, §8]. Oneproperty that will be useful below is that W has a left-derivative on (0 , ∞ ), which wewill write as W ′ .Having recalled some of the theory of scale functions, we are in a position to identifythe invariant measure we seek. Proposition 10.
Suppose we are in the case (PR) or (NR) . Let m be the pushforwardof the measure W (d x ) by the function x ce − x , that is, m (d y ) = 1 a δ c (d y ) + W ′ (log( c/y )) d yy . Then, E c hR H ( c )0 f ( Y s ) d s i = a B h f, m i .In case (PR) , h , m i = (cid:16) a + R log s ρ (d s ) (cid:17) − , and ν = m/ h , m i is the invariantdistribution of Y .In case (NR) , h , m i = ∞ , and m is a (unique up to a multiplicative constant) σ -finiteinvariant measure of Y . In the latter case, define ν = m .Proof. We give quite a general proof, which works under assumptions (PR) and (NR),and even under rather general assumptions about Y (or η ). In the case (PR) and underour running assumption that η has no Gaussian part and finite jump rate, a simpler (ifnot shorter) proof is available; see Remark 11(i).Let R t = ¯ η t − η t , the process η reflected in its running maximum, and define L − t :=inf { s > η > t } to be the inverse local time of η at the maximum. Let P = P c . Under P , we have Y t = e b − R t , so L − can be interpreted either as the inverse local time of R at 0 or the inverse local time of Y at c .For arbitrary f , define g ( x ) = f ( ce − x ). Following the proof of [7, Theorem VI.20], weobtain E Z L − t f ( Y s ) d s = E Z L − t g ( R s ) d s = E Z ∞ g ( R s ) { ¯ η s
0, using the same method as [33,Theorem 8.7], and then letting x → ∞ . Then, the invariance of m follows, underassumption (PR), from [1, VI.1.5].(ii) We also note that m is still an invariant measure when ξ (or η ) is replaced by anarbitrary spectrally negative Lévy process, even if it has a Gaussian part or infinitejump rate; however, then m should be interpreted in terms of the excursions of Y from c .(iii) The measure W has Laplace transform β βψ η ( β ) . Under assumption (PR), since ν is the unique invariant probability measure for Y , the previous result is equivalent,by Laplace inversion, to [1, Corollary IX.3.4], which gives another proof even underthe general Lévy assumption. However, we are not aware of an existing proof ofthis result in the null recurrent case.We are now in a position to solve the eigenproblem (4) and state the asynchronousexponential growth behaviour (6), which are interesting results in their own right, as wellas stepping stones in our proof of the strong law of large numbers. It is worth pointingout that, whereas our main strong law result, Theorem 1, is stated under the assumptionthat λ ∗ > roposition 12. Assume (NR) or (PR) . Let h ≡ and ν be as defined above. Then:(i) e λt is the spectral radius of Ψ t .(ii) h ∈ D ( A ) and A h = λh .(iii) For every f ∈ D ( A ) ∩ L ∞ (0 , c ] , ν A f = λνf .Proof. (i) This follows easily from the fact that λ = t log Ψ t f , for every boundedfunction f .(ii) The first claim follows from Corollary 5 and the second from the definition of λ .(iii) First note that from the proof of Proposition 4 and Corollary 5, the set of function D is a core for ( A , D ( A )). Hence, we first show that this claim holds for f ∈ D ∩ L ∞ (0 , c ].Due to Proposition 6 and the fact that ν is the left eigenmeasure for Ψ t corres-ponding to the eigenvalue e λt , we have ν Ψ t f = νW t f . Then, due to Corollary 5,the claim follows for f ∈ D ∩ L ∞ (0 , c ].Now choose f ∈ D ( A ) ∩ L ∞ (0 , c ]. Since D is a core of A , this means we canchoose a sequence of functions ( f n ) n ≥ ⊂ D such that k f n − f k A → n → ∞ ,where k · k A denotes the graph norm. In particular, this means that both f n → f and A f n → A f in L ((0 , c ]) as n → ∞ , and hence the result follows. Proposition 13.
Assume (PR) . For all f ∈ C b ((0 , c ]) , and all x ∈ (0 , c ] , lim t →∞ e − λt Ψ t f ( x ) = h f, ν i . If (ER) holds, then there exists k > such that for all x ∈ (0 , c ] and f ∈ C b ((0 , c ]) , lim t →∞ e kt (cid:12)(cid:12) e − λt Ψ t f ( x ) − h f, ν i (cid:12)(cid:12) = 0 . Proof.
This is a corollary of Proposition 8 and Proposition 10.The two preceding propositions prove equations (4) and (6) in the introduction; thatis, they demonstrate the asynchronous exponential growth of the expectation semigroup,and its connection with the spectral theory of the operator A .
5. Law of large numbers
This section is dedicated to the proof of Theorem 1, so we assume the process is super-critical (i.e., λ ∗ = B − k >
0) and Y is exponentially recurrent. Recall that in this case λ = λ ∗ .Let M t = e − λt N t , N t is the number of cells alive at time t . Then, M is a martingale; indeed, it isthe intrinsic martingale in a continuous-time branching process (regardless of x ). By [2,§III.4, p. 108], E x [ M t ] = 2(1 − e − λt ), so M is L -bounded, and hence has a limit M ∞ almost surely and in L . Note that Doob’s martingale inequality also yields convergencein L . A key element in the proof is a skeleton decomposition for the growth-fragmentationprocess. This corresponds to splitting Z into a tree of particles that survive forever,dressed with trees of particles that all die out.In order to describe this precisely, we start by specifying the non-linear semigroupof the growth-fragmentation process; this describes the behaviour of all the particles,rather than the average characterised by Ψ t .Recall U t , the set of labels of particles alive at time t , and for measurable functions f with k f k ∞ <
1, define the non-linear semigroup u t [ f ]( x ) := E x Y u ∈U t f ( Z u ( t )) , t ≥ , x ∈ (0 , c ] . Proposition 14.
The semigroup ( u t ) t ≥ satisfies the following non-linear growth-fragmentationequation u t [ f ]( x ) = f ( xe a · t ∧ T c ( x ) ) + Z t G [ u t − s [ f ]]( xe a · s ∧ T c ( x ) )d s, (12) where G [ f ]( x ) = ( B + k ) (cid:18) BB + k Z d vκ ( v ) f ( xv ) f ( x (1 − v )) + k B + k − f ( x ) (cid:19) (13) and κ ( v ) = π ( v ) /vB .Proof. Conditioning on the time of the first event (fragmentation or killing), we have u t [ f ]( x ) = f ( xe a · t ∧ T c ( x ) ) e − ( B + k ) t + Z t d s ( B + k ) e − ( B + k ) s × (cid:18) BB + k Z −∞ Π(d u ) B u t − s [ f ]( e a · s ∧ T c ( x ) e u ) u t − s [ f ]( e a · s ∧ T c ( x ) (1 − e u )) + k B + k (cid:19) . Following the same steps as the linear case, namely making the change of variables v = e u and applying Dynkin’s lemma, we obtain the required result.For ease of notation, we will henceforth write G [ f ]( x ) = ( B + k ) E [ f ( xV ) f ( x (1 − V )) N =2 + N =0 − f ( x )] , where under P with expectation E , N ∈ { , } is the number of particles produced atthe first event (fragmentation or killing) with P ( N = 2) = B/ ( B + k ), and V is chosenaccording to κ . 19e turn now to the skeleton decomposition. Let C u denote the ‘colour’ of cell u . Wewrite C u = b if the descendants of cell u survive forever (a ‘blue’ particle) and C u = r ifthe descendants of cell u eventually become extinct (a ‘red’ particle). Let C ( t ) = ( C u : u ∈ U t ).Define ζ := inf { t ≥ N t = 0 } to be the lifetime of the process, p := P x ( ζ = ∞ ) tobe the survival probability and w = 1 − p to be the extinction probability. Note thatsince the fragmentation kernel and rate do not depend on the particle size, p and w donot depend on x ∈ (0 , c ]. By Proposition 7, w = k B < Lemma 15.
Given F t , the colours of the cells of Z ( t ) are given by independently choos-ing each cell to be blue with probability p and red with probability w .Proof. This is shown in [30]. The independence of the colours is due to the spatialhomogeneity of the branching and killing rates.We denote by ( Z , C ) = (( Z ( t ) , C ( t )) : t ≥
0) the process of cells marked by theircolours.We first describe the ‘red trees’. For A ∈ F t define P Rx ( A ) := P x ( A | C ∅ = r ) . (14) Proposition 16 (Red trees) . The dynamics of the process ( Z , P Rx ) can be described asfollows. From a single particle at position x ∈ (0 , c ] , the particle will grow according to s xe a · s ∧ T c ( x ) , as before. Then, for x ∈ (0 , c ] and bounded, measurable f , the branchinggenerator is given by G R [ f ]( x ) = 1 w ( G [ f w ]( x ) − f ( x ) G [ w ]( x )) . (15)= ( B + k ) E (cid:20) k B + k f ( xV ) f ( x (1 − V )) + BB + k − f ( x ) (cid:21) The proof of this proposition can be found in [30, Prop 2.1]. We now consider thecase where we condition on C ∅ = b . In order to do so, we first define a process ( Z , P B ),which should be thought of as a process representing the blue cells only. Started froman initial cell of size x ∈ (0 , c ], the cell increases in size according to s xe a · s ∧ T c ( x ) andwill fragment according to the following operator, G B [ f ]( x ) = 1 p ( G [ pf + w ] − (1 − f ) G [ w ]) (16)= B ( p + 2 w ) E (cid:20) pp + 2 w f ( xV ) f ( x (1 − V ) + wp + 2 w (cid:0) f ( xV ) + f ( x (1 − V )) (cid:1) − f ( x ) (cid:21) Note B ( p + 2 w ) = B + k .We are now ready describe the skeleton decomposition.20 roposition 17 (Skeleton decomposition) . With probability w , ( Z , C , P x ) is equal inlaw to ( Z , P Rx ) with all cells coloured red; and with probability p , ( Z , C , P x ) is equal inlaw to ( Z , C , P x ( · | C ∅ = b )) , which, in turn, is equal in law to a process ( Z , P Bx ) colouredblue, dressed with copies of ( Z , P Rx ) coloured red. In the latter case, the joint branchingof the blue tree and dressing with red trees can be described via the following generator: H [ f, g ]( x ) = B (2 w + p ) E [ p w + p f ( xV ) f ( x (1 − V ))+ w w + p f ( xV ) g ( x (1 − V )) + w w + p g ( xV ) f ( x (1 − V ) − f ( x )] (17)The idea behind this proposition is that each particle in the growth-fragmentationprocess can either be coloured red, if its genealogy goes extinct, or blue, if its genealogysurvives forever. In the case where the initial particle is blue, the branching operator H describes the colours of the offspring when a fragmentation occurs. In particular,the production of blue particles is described by the function f , whereas g describesthe production of red particles. We can see from (17) that the first summand in theexpectation on the righthand side of (17) describes the case where a particle fragmentsand both fragments are blue. The second (resp. third) summand then describes the casewhere the fragment of size xV (resp. x (1 − V )) is blue and the other is red.The proof of this proposition follows from [30, Section 2.3] by setting ς ( x ) = B + k andusing the specific form of the branching generator given in (13). In particular, the readermay find the proof for the exact formulation of H in the proof of [30, Proposition].In order to prove the strong law, we will first prove it for the blue process ( Z , P B ),and then show that this implies the same result for ( Z , P ).The combination of Lemma 15 and Proposition 17 gives the following identity, whichwill be very useful. E Bx [ h f, Z ( t ) i ] = E x "X u f ( Z u ( t )) { C u = b } (cid:12)(cid:12)(cid:12)(cid:12) C ∅ = b = 1 p E x "X u f ( Z u ( t )) { C u = b } { C ∅ = b } = 1 p E x "X u f ( Z u ( t )) { C u = b } = 1 p E x [ h f, p Z ( t ) i ]= E x [ h f, Z ( t ) i ] , (18)where the first equality follows from the proposition; the third equality uses the fact thatthe sum is empty if the initial particle is not blue; and the penultimate equality comesfrom the lemma.An immediate consequence is that if ( λ ∗ , , ν ) is the eigen-triple for ( Z , P ), then( λ ∗ , /p, pν ) is the eigen-triple for the blue process ( Z , P B ). Moreover, the process Y defined by E Bx [ f ( Y t )] = e − λ ∗ t E Bx [ h f, Z ( t ) i ] is the same in distribution as the process Y we defined earlier for ( Z , P ). 21nder the measures P B , we retain our notation N t for the number of blue particlesalive at time t ≥
0. Since λ is still the leading eigenvalue for the blue process, it followsthat M t := e − λt N t is a positive martingale under P Bx , for each x ∈ (0 , c ]. Let M ∞ denote its limit. We willshow in what follows that M is L ( P Bx )-convergent.Our intermediate strong law is as follows. Theorem 18.
For all x ∈ (0 , ∞ ) and continuous f with k f k < , e − λt h f, Z ( t ) i → h f, ν i M ∞ , P Bx -almost surely and in L ( P Bx ) . In this part, we follow closely the ideas of [26], though of course our processes are verydifferent in nature.
Lemma 19.
Let U t = e − λ ∗ t h f, Z ( t ) i . Then, for any increasing sequence ( m n ) n ≥ , U ( m n + n ) δ − E Bx [ U ( m n + n ) δ | F nδ ] → , as n → ∞ , P Bx -a.s. and in L ( P Bx ) .Proof. For convenience, we will write m instead of m n . For almost sure convergence, bythe Borel-Cantelli lemma, it is sufficient to show that, for all ǫ > X n ≥ P Bx (cid:16)(cid:12)(cid:12) U ( m + n ) δ − E Bx [ U ( m + n ) δ | F nδ ] (cid:12)(cid:12) > ǫ (cid:17) < ∞ . We first note that E Bx "(cid:12)(cid:12)(cid:12)(cid:12) U ( m + n ) δ − E Bx [ U ( m + n ) δ | F nδ ] (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) F nδ = E Bx "(cid:12)(cid:12)(cid:12)(cid:12) X u e − nδλ (cid:0) U ( u ) mδ − E BZ u ( nδ ) [ U ( u ) mδ ] (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F nδ , where, conditional on F nδ , the U ( u ) mδ are independent and distributed as ( U, P Z u ( nδ ) ) for u ∈ U nδ . It follows that E Bx "(cid:12)(cid:12)(cid:12)(cid:12) U ( m + n ) δ − E Bx [ U ( m + n ) δ | F nδ ] (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) F nδ = E Bx "(cid:12)(cid:12)(cid:12)(cid:12) X u e − nδλ (cid:0) U ( u ) mδ − E BZ u ( nδ ) [ U ( u ) mδ ] (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F nδ = E Bx (cid:20)X u e − nδλ (cid:12)(cid:12) U ( u ) mδ − E BZ u ( nδ ) [ U ( u ) mδ ] (cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) F nδ (cid:21) ≤ E Bx "X u e − nδλ (cid:16)(cid:12)(cid:12) U ( u ) mδ (cid:12)(cid:12) + (cid:12)(cid:12) E BZ u ( nδ ) [ U ( u ) mδ ] (cid:12)(cid:12) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) F nδ ≤ e − λnδ X u E BZ u ( nδ ) (cid:20)(cid:16) U ( u ) mδ (cid:17) (cid:21) ≤ e − nδλ k f k sup x,t E Bx [( M t ) ] · M nδ . (19)22here, for the second equality, we used that the sum is over conditionally independent,zero-mean summands, for the first inequality we have used that | a + b | ≤ | a | + | b | )and for the second inequality we have used Jensen’s inequality.A similar calculation to that of (18) shows that E Bx [( M t ) ] = p E x [ M t ], and hence, M is L ( P Bx )-bounded, uniformly in x ∈ (0 , c ]. From Doob’s inequality, it follows that M is L ( P Bx )-convergent and hence L ( P Bx )-convergent. Therefore, E Bx h(cid:0) U ( m + n ) δ − E Bx [ U ( m + n ) δ | F nδ ] (cid:1) i ≤ k f k sup x,t E Bx [ M t ] · e − nδλ , which is summable, so the Markov inequality completes the proof of the almost sureconvergence. Lemma 20.
Theorem 18 holds when restricted to lattice times: lim n →∞ e − nδλ h f, Z ( nδλ ) i = h f, ν i M ∞ , P Bx -almost surely and in L ( P Bx ) .Proof. Noting that E Bx (cid:2) e − λ nδ h f, Z (2 nδ ) i (cid:12)(cid:12) F nδ (cid:3) = e − λnδ X u e − λnδ E BZ u ( nδ ) [ h f, Z ( nδ ) i ] , we have e − λ nδ h f, Z (2 nδ ) i = e − λ nδ h f, Z (2 nδ ) i − e − λ nδ E Bx (cid:2) h f, Z (2 nδ ) i (cid:12)(cid:12) F nδ (cid:3) + e − λnδ X u (cid:16) e − λnδ E BZ u ( nδ ) (cid:2) h f, Z ( nδ ) i (cid:3) − h f, ν i (cid:17) + h f, ν i M nδ . The first term goes to zero a.s. and in L ( P Bx ), by setting m n = n in the precedinglemma. The last term approaches h f, ν i M ∞ by the martingale convergence discussedpreviously.If we define R s,t = X u e − λt | e − λs E BZ u ( t ) [ h f, Z ( s ) i ] − h f, ν i| , then, using again (18), we have R s,t = X u e − λt | e − λs E BZ u ( t ) [ h f, Z ( s ) i ] − h f, ν i| = X u e − λt | E BZ u ( t ) [ f ( Y s )] − h f, ν i| it is sufficient to show that lim n →∞ R nδ,nδ = 0 almost surely. We now wish to applyProposition 8. This was proven under the measures P · , but as remarked above, thedistribution of the spine Y is the same under P B · . By Proposition 8, choose w ∈ (0 , r )such that ψ η ( − w ) <
0; then, there exist k >
C > x ∈ (0 , c ] and t ≥ (cid:12)(cid:12) E Bx [ f ( Y s )] − h f, ν i (cid:12)(cid:12) ≤ k f w k (cid:0) ( c/x ) w + C (cid:1) e − ks , f w ( x ) = ( x/c ) w f ( x ).Applying this in our situation, we have E x [ R s,t ] ≤ e − ks k f w k E x (cid:2) ( c/Y t ) w + C (cid:3) = e − ks k f w k E x [ e w ( b − η bt ) + C ] . We estimate: E x [ e w ( b − η bt ) ] = E x [ e w (¯ η t ∨ b − η t ) ] ≤ E x [ e w ( b +¯ η t − η t ) ]= E [ e w ( b +¯ η t − η t ) ]= e wb E [ e − wη t ] ≤ e wb E [ e − wη ∞ ] . We will show that the right-hand side is finite, using the Wiener-Hopf factorisation of η .Let ˆ H denote the descending ladder height process of η and ˆ κ (0 , · ) its Laplace expo-nent. By [33, p. 178], we know that η ∞ d = − ˆ H e χ , where ˆ H is an unkilled version of ˆ H ,and e χ is an independent exponential random variable with rate χ = ˆ κ (0 , > e χ , we see that E [ e w ˆ H e χ ] < ∞ if and only if E [ e w ˆ H ] < ∞ and ˆ κ (0 , − w ) >
0. Furthermore, by [33, Theorem 7.8], weknow that the Lévy measures of ˆ H and η are related by Π ˆ H (d y ) = Π η ( y, ∞ ) d y , so if E [ e wη ] < ∞ , (20)then E [ e w ˆ H ] < ∞ also.Now, by assumption, our choice of w satisfies (20) and ψ η ( − w ) <
0, so the results of[33, §6.5.2] imply that ˆ κ (0 , − w ) = ψ η ( − w ) − w >
0. We conclude that E [ e − wη ∞ ] < ∞ .It follows that E x [ R s,t ] ≤ e − ks k f w k C ′ , with C ′ = e wb E [ e − wη ∞ ] + C . Hence, we have convergence in L . Finally, by the Markovproperty, X n ≥ P x ( R nδ,nδ > ǫ ) < ∞ . The Borel-Cantelli lemma implies that lim n →∞ R nδ,nδ = 0 almost surely. Proof of Theorem 18.
To complete the proof of Theorem 18, we need to pass from latticetime to continuous time. For the L ( P Bx ) convergence, the Croft-Kingman lemma yieldsthe result. For the almost sure part, the proof is identical to the proof given in [26], andso we omit it. In this section, we work under the probability measure P , and will make use of both thefull growth-fragmentation process Z and the part of it coloured blue, which we denoteby Z B (and which, if it exists, has the law of ( Z , P B )). The intrinsic martingale of theblue tree will be denoted M B . 24irst note that the L convergence for the full process follows easily from the relation(18). For the almost sure convergence, the proof follows the same idea as [30, §4], usingthe following proposition, a discussion of whose proof can also be found in that reference. Proposition 21.
Let (Ω , F , ( F t , t ≥ , P ) be a filtered probability space and define F ∞ := σ ( ∪ ∞ i =1 F t ) . Suppose ( U t , t ≥ is an F -measurable non-negative process suchthat sup t ≥ U t has finite expectation and ( E ( U t |F t ) , t ≥ is càdlàg. If lim t →∞ E ( U t |F ∞ ) = Y, a.s,then lim t →∞ E ( U t |F t ) = Y, a.s. . To put the above proposition in the context of the growth-fragmentation setting,set U t = e − λt h f, Z B ( t ) i , for f satisfying the conditions of Theorem 1, and recall that( F t , t ≥
0) is the filtration generated by the growth-fragmentation process ( Z ( t ) , t ≥ U t , t ≥
0) by a multiple of ( M Bt , t ≥
0) and hence weautomatically get that sup t ≥ U t has a second, and hence first, moments thanks to (18).Due to Theorem 18 and the fact that Z B ( t ) is F ∞ -measurable, U t = E ( U t | F ∞ ) andhence lim t →∞ E ( U t | F ∞ ) = h f, ν i M B ∞ , P x -almost surely, for x ∈ (0 , c ].Using (18), we get E ( U t | F t ) = E ( e − λt h f, Z B ( t ) i| F t ) = e − λt p h f, Z ( t ) i . Combining this with Proposition 21 yieldslim t →∞ e − λt h f, X t i = h f, ν i M B ∞ /p, (21) P x -almost surely.To complete the proof of almost sure convergence, we need to show that M B ∞ /p = M ∞ ,almost surely. To do so, take f = 1 in (21) and observe that the left-hand side is M ∞ .
6. Long-term behaviour in the transient regime
Finally, in this last section, we will prove Theorem 2. Consider the case where Z issupercritical but (T) holds, so that λ = q ∗ = λ ∗ + inf ψ η < λ ∗ .To simplify the exposition, in this section we will assume that k = 0, that is, thatthere is no killing of cells. In the general case, the results of this section can be provedusing the skeleton decomposition, as in the previous section.Now, since we are assuming that (T) holds, we no longer have the advantage of beingable to use M t to define a change of measure, as in the previous section. However, inthis case, we are able to use the function ℓ ( x ) = L x,c ( λ ) = ( x/c ) arg inf ψ η to yield a useful supermartingale change of measure.Our techniques in this section are inspired by a combination of [11] and [26].25 roposition 22. (i) The process S t = e − ( λ − λ ∗ ) t ℓ ( Y t ) ℓ ( x ) = e − inf ψ η · t ℓ ( Y t ) ℓ ( x ) is a P x -supermartingalefor the natural filtration F Y of Y . Under the change of measure d˜ P x d P x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F Yt = e − inf ψ η · t ℓ ( Y t ) ℓ ( x ) ,Y is the exponential of a Lévy process with Laplace exponent ˜ ψ η ( q ) = ψ η ( q +arg inf ψ η ) − inf ψ η , reflected in the level c and killed according to the multiplicativefunctional t e − a · arg inf ψ η · R t { Ys = c } d s .(ii) The process S t = e − λt ℓ ( x ) h ℓ, Z i is a P x -supermartingale for the natural filtration of Z . A measure ˜ P x supporting Z and the additional random variable ζ can be definedas follows: ˜ P x ( F t { ζ>t } ) = P x ( F t S t ) , F t ∈ F t . Under this measure, the process Z has the following decomposition. There is asingle distinguished cell whose mass has the distribution of the process Y under ˜ P x ,and at every jump ∆ Y t of this cell, a copy of Z under measure P − ∆ Y t is introduced,which we can denote Z [ t ] . That is, Z has the same distribution as the process t δ Y t + X . Then, sup x ∈ (0 ,c ] sup t ≥ ℓ ( x ) E x [ S t ] < ∞ .Proof. (i) Let ˆ ℓ ( y ) = e arg inf ψ η · ( y − b ) , so that ℓ ( Y t ) = ˆ ℓ ( η bt ).The P x -supermartingale property of S is equivalent to the statement that E y [ e − inf ψ η · t ˆ ℓ ( η bt ) / ˆ ℓ ( x )] ≤ , since η b is Markov (where y = log x .) Recall that E y [ e αη t − ψ η ( α ) t − αy ] = 1, for any α ≥
0. Then, E y [ e αη bt − ψ ( α ) t − αy ] = e − αy − ψ η ( α ) t E y [ e α ( η t − (¯ η t − b ) ∨ ]= E y [ e − α (¯ η t − b ) ∨ ] ≤ . Setting α = arg inf ψ η > Y . It appears to be simplest to demonstratethis using evolution equations. Therefore, let (Φ t ) t ≥ represent the semigroup of Y under P · , and ( ˜Φ t ) t ≥ the same object under ˜ P · .It is simple to show, using the same ideas as in section 2, that Φ satisfies theevolution equationΦ t f ( x ) = f ( xe a · t ∧ T c ( x ) )+ Z t d s Z (Π η ◦ exp − )(d v ) (cid:2) Φ t − s f ( xe a · s ∧ T c ( x ) v ) − Φ t − s f ( xe a · s ∧ T c ( x ) ) (cid:3) . t f ( x ) = e − inf ψ η · t ℓ ( x ) Φ t ( f ℓ )( x ) . Rewriting the above evolution equation in terms of ˜Φ, performing some algebraicmanipulation and making use of Dynkin’s integral identity (Lemma 27) we obtainthe following evolution equation for ˜Φ:˜Φ t f ( x ) = f ( xe a · t ∧ T c ( x ) ) − Z tT c ( x ) αa ˜Φ t − s f ( xe a · s ∧ T c ( x ) ) d s + Z t d s Z ( ˜Π η ◦ exp − )(d v ) (cid:2) ˜Φ t − s f ( xe a · s ∧ T c ( x ) v ) − ˜Φ t − s f ( xe a · s ∧ T c ( x ) ) (cid:3) , where α = arg inf ψ η and ˜Π η is the Lévy measure of the Lévy process with Laplaceexponent ˜ ψ η given in the statement. It is clear that this is the evolution equationassociated with the process described in the statement of the result.It remains to show that the above evolution equation characterises ˜Φ. Since thekilling and jump rates are bounded, this follows using Gronwall’s inequality exactlyas in section 2.(ii) This proof is very similar to the classical methods of [29], and we give only anoutline. The first step is to introduce an additional (killed) process I t on U bydefining ˜ P x ( F t { I t = u } ) = e − λt ℓ ( x ) P x ( F t ℓ ( Z u ( t ))) , F t ∈ F t , and declaring ζ to be the killing time of I . This random variable I t indicates theindex of the distinguished ‘spine’ cell.It follows that˜ E x [ f ( Z I s ( s ) , s ≤ t )] = e − λt ℓ ( x ) X u E x [ f ( Z u ( s ) , s ≤ t ) ℓ ( t )] , and in particular, taking part (i) into account, we see that under ˜ P x , t Z I t ( t )has the same distribution as the process Y .Denote by T k ( u ) the k -th jump of the cell labelled u (or its ancestors), and by T k the k -th jump of Y . Since all processes involved are Markov, it suffices tocheck the decomposition at fixed times, and we will focus first on the case where T ( I t ) ≤ t < T ( I t ); that is, between the first and second jumps of the spine.We define an operator r on U which removes the prefix, i.e., if u = u u u · · · ,27 u = u u · · · . Now let f and g be measurable functions.˜ E x h f ( Z I t ( t )) g ( Z v ( t ) , v = I t ) { ζ>t } { T ( I t ) ≤ t
Let f be such that k f /ℓ k ∞ < ∞ , and fix m ≥ . Define U t = e − λt h f, Z ( t ) i .Then, U ( m + n ) δ − E x [ U ( m + n ) δ | F nδ ] → , as n → ∞ , P x -a.s. and in L ( P x ) .Proof. To be concise, let s = mδ and t = nδ . Using the conditional independence of thezero-mean summands, we get E x h ( U s + t − E x [ U s + t | F t ]) (cid:12)(cid:12)(cid:12) F t i = E x " e − λt X i (cid:16) e − λs h f, Z ( i ) ( s ) i − E Z i ( t ) [ e − λs h f, Z ( s ) i ] (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t = X i e − λt (cid:16) E Z i ( t ) [( e − λs h f, Z ( s ) i ) ] − E Z i ( t ) [ e − λs h f, Z ( s ) i ] (cid:17) ≤ k f /ℓ k ∞ X i e − λt E Z i ( t ) [( e − λs h ℓ, Z ( s ) i ) ] ≤ k f /ℓ k ∞ X i e − λt ℓ ( Z i ( t )) E Z i ( t ) [ S s ] ≤ k f /ℓ k ∞ · sup y,u ℓ ( y ) E y [ S u ] · e − λt ℓ ( x ) S t . Hence, using that S is a supermartingale, E x h ( U s + t − E x [ U s + t | F t ]) i ≤ const · e − λt ℓ ( x ) . Hence, the L ( P x )-convergence to zero holds. Moreover, this is summable in n (recallthat t = nδ ) and so, using an application of Markov’s inequality and the Borel-Cantellilemma, the almost sure convergence to zero holds too.We are now able to state and prove the following rephrasing of our second maintheorem. To understand the connection with the statement in the introduction, whichis in terms of the ‘cumulant’ κ , observe that κ (0) = λ ∗ and that ψ η ( q ) = κ ( q ) − κ (0) . Hence, inf κ = inf ψ η + λ ∗ and arg inf κ = arg inf ψ η , so that in particular λ = inf κ .29 heorem 25 (Rephrasing of Theorem 2) . Assume that λ > and that f : (0 , c ] → R iscontinuous and bounded with f ( x ) = O ( x arg inf ψ η ) as x → . Then, e − λt h f, Z ( t ) i → , P x -almost surely and in L ( P x ) .Proof. We begin with the proof for lattice times, and for simplicity, we assume f ≥ δ >
0, we have, as in the recurrent case, e − λ ( s + t ) h f, Z ( s + t ) i = e − λ ( s + t ) h f, Z ( s + t ) i − E x (cid:2) e − λ ( s + t ) h f, Z ( s + t ) i (cid:12)(cid:12) F t (cid:3) + e − λt X i E Z i ( t ) (cid:2) e − λs h f, Z ( s ) i (cid:3) . The first term converges to zero in the sense stated along lattice times, by the precedinglemma. The second term can be expressed X i e − λt E Z i ( t ) (cid:2) e − λs h f, Z ( s ) i (cid:3) = X i e − λt ℓ ( Z i ( t ))˜ E Z i ( t ) (cid:20) f ( Y s ) ℓ ( Y s ) (cid:21) ≤ k f /ℓ k ∞ S t . Since S is a positive supermartingale, it converges almost surely [39, Corollary II.2.11].Moreover, E x [ S t ] = ˜ P x ( ζ > t ) →
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We make use of the following identity in the proof of Proposition 22. It is an adaptationof [24, §4, Lemma 1.2] to the context of inhomogeneous killing (or branching) rates.
Lemma 27.
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