Self-similar solutions of fragmentation equations revisited
aa r X i v : . [ m a t h . A P ] F e b SELF-SIMILAR SOLUTIONS OF FRAGMENTATIONEQUATIONS REVISITED
WERONIKA BIEDRZYCKA AND MARTA TYRAN-KAMI ´NSKA
Abstract.
We study the large time behaviour of the mass (size) ofparticles described by the fragmentation equation with homogeneousbreakup kernel. We give necessary and sufficient conditions for the con-vergence of solutions to the unique self-similar solution. Introduction
Fragmentation is a phenomenon of breaking up particles into a rangeof smaller sized particles, characteristic of many natural processes rangingfrom e.g. polymer degradation [33] in chemistry to breakage of aggregates[1] in biology. A stochastic model of fragmentation was given in [16] for thefirst time and since then it has been studied extensively with probabilisticmethods, see [8, 19, 30, 31] and the references therein. There is also adeterministic approach through transport equations and purely functional-analytic methods [23, 22, 6, 2, 15, 5, 25], which we follow here. We denoteby c ( t, x ) the number density of particles of mass (size) x > t > ∂c ( t, x ) ∂t = Z ∞ x b ( x, y ) a ( y ) c ( t, y ) dy − a ( x ) c ( t, x ) , t, x > , (1)with initial condition c (0 , x ) = c ( x ) , x > , (2)where a ( x ) is the breakage rate for particles of mass x and b ( x, y ) is theproduction rate of particles of size x from those of size y . Both a and b arenonnegative Borel measurable functions. To ensure the conservation of thetotal mass the function b has to satisfy Z y b ( x, y ) xdx = y and b ( x, y ) = 0 for x ≥ y. If we let γ ( y, x ) = b ( x, y ) a ( y ) then (1) has the same form as in [23], wherethe reader can find a brief physical interpretation and the derivation of theequation. For a discussion of the model we also refer to [5, Chapter 8]. Date : September 18, 2018.2010
Mathematics Subject Classification.
Primary 47D06; Secondary 60J25, 60J35,60J75.
Key words and phrases. piecewise deterministic Markov process, substochastic semi-group, strongly stable semigroup, fragmentation models.This research was supported by the Polish NCN grant No. 2014/13/B/ST1/00224.
In this paper we provide necessary and sufficient conditions for the ex-istence of self-similar solutions to the initial value problem (1)–(2) when a ( x ) = x α with α > b is homogenous , i.e., there exists aBorel measurable function h : (0 , → R + such that b ( x, y ) = 1 y h (cid:18) xy (cid:19) for 0 < x < y and Z h ( r ) rdr = 1 . (3)This means that the size x of fragments of particles is proportional to the size y of fragmenting particles, so that it is determined through the distributionof the ratio x/y and does not depend on y . We let E denote the openinterval (0 , ∞ ), E = B (0 , ∞ ) be the σ -algebra of Borel subsets of (0 , ∞ ), and L = L ( E, E , m ) be the space of functions integrable with respect to themeasure m ( dx ) = x dx with the norm k f k = Z ∞ | f ( x ) | xdx. If a ( x ) = x α with α ≥ c ∈ D ( m ) := { f ∈ L : f ≥ , k f k = 1 } then c ( t, · ) ∈ D ( m ) for all t >
0, and if α < α > c of the fragmentation equation (1) as c ( t, x ) = γ ( t ) u (log γ ( t ) , γ ( t ) x ) , x, t > , where γ ( t ) = (1 + t ) /α and u is the solution of the following partial integro-differential equation ∂u ( t, x ) ∂t = − x ∂∂x ( x u ( t, x )) − ϕ ( x ) u ( t, x ) + P ( ϕu ( t, · ))( x ) . (4)Here the function ϕ and the linear operator P on L are defined as ϕ ( x ) = αx α and P f ( x ) = Z ∞ x y h (cid:16) xy (cid:17) f ( y ) dy, x > . (5)In particular, if u ∗ is a stationary solution of (4) then c ∗ ( t, x ) = γ ( t ) u ∗ ( γ ( t ) x ) , t ≥ , x > , γ ( t ) = (1 + t ) /α , (6)is said to be a self-similar solution of (1). It should be noted that c behavesas a delta-like function; see Remark 2.Our main result is the following. Theorem 1.
Let a ( x ) = x α with α > and let b be as in (3) . There existsa self-similar solution c ∗ as in (6) with c ∗ ( t, · ) ∈ D ( m ) , t ≥ , and everysolution c of (1) with initial condition c ∈ D ( m ) satisfies lim t →∞ Z ∞ | c ( t, x ) − c ∗ ( t, x ) | x dx = 0 if and only if Z log z h ( z ) zdz > −∞ . (7) ELF-SIMILAR SOLUTIONS OF FRAGMENTATION EQUATIONS REVISITED 3
To our knowledge this result is new in the given generality. Particularself-similar solutions were obtained in [26, 32] using various methods. For aprobabilistic approach to self-similar fragmentation we refer the reader to [8]and the references therein. In the mathematical literature, the self-similarsolutions and the asymptotic behaviour for fragmentation equation has beenconsidered using deterministic analytic methods in [15] where the existenceof self-similar solution is proved under the assumption that Z z k h ( z ) dz < ∞ for some k < k ≤ α with additionalregularity constraints. The proof of our result is based on the representa-tion of solutions of (1) as densities of a Markov process { Y ( t ) } t ≥ ; see (12)and (14). We define another Markov process { X ( t ) } t ≥ corresponding to thegrowth-fragmentation equation (4) such that Y ( t ) = (1 + t ) − /α X (log(1 + t ) /α ) for all t ≥ . (8)This is a piecewise deterministic Markov process with good asymptotic be-haviour, as will be shown in Section 3 using results from [11] based onproperties of stochastic semigroups. In Lemma 4 we give a relation betweencondition (7) and the first jump time of the process { X ( t ) } t ≥ . At the endof Section 3 we also define such processes in the case when α < α = 0 is treated in [14] and[10]. If α > { /Y ( t ) } t ≥ is an ex-ample of a so-called self-similar Markov processes with increasing samplepaths and it follows from [9, Theorem 1] that t /α Y ( t ) converges in distribu-tion to a random variable Y ∞ , which is non-degenerate under condition (7).Our approach provides convergence of densities of (1 + t ) /α Y ( t ), implyingconvergence in distribution and identifies the distribution of Y ∞ as being astationary distribution of the process { X ( t ) } t ≥ , absolutely continuous withrespect to m and with density u ∗ . If (7) does not hold then Y ∞ is equal to 0.At the end of Section 4 we describe how to get this type of limit using ourapproach. Further study of asymptotic behaviour in this case is providedin [12]. Self-similar Markov processes were also used in [19] to get the largetime behaviour of solutions of the fragmentation equation when α < Preliminaries
Let ( E, E , m ) be a σ -finite measure space and L = L ( E, E , m ) be thespace of integrable functions. We denote by D ( m ) ⊂ L the set of all densi-ties on E , i.e. D ( m ) = { f ∈ L : k f k = 1 } , where L = { f ∈ L : f ≥ } , WERONIKA BIEDRZYCKA AND MARTA TYRAN-KAMI ´NSKA and k · k is the norm in L . A linear operator P : L → L such that P ( D ( m )) ⊆ D ( m ) is called stochastic or Markov [20]. It is called sub-stochastic if P is a positive contraction, i.e., P f ≥ k P f k ≤ k f k for all f ∈ L .Let P : E × E → [0 ,
1] be a stochastic transition kernel , i.e., P ( x, · ) isa probability measure for each x ∈ E and the function x
7→ P ( x, B ) ismeasurable for each B ∈ E , and let P be a stochastic operator on L . If Z E P ( x, B ) f ( x ) m ( dx ) = Z B P f ( x ) m ( dx ) (9)for all B ∈ E , f ∈ D ( m ), then P is called the transition operator correspond-ing to P . Suppose that there exists a measurable function p : E × E → [0 , ∞ )such that P f ( x ) ≥ Z E p ( x, y ) f ( y ) m ( dy )for m -a.e. x ∈ E and for every density f . If p can be chosen in such a waythat Z E Z E p ( x, y ) m ( dx ) m ( dy ) > P is called partially integral and if p is such that Z E p ( x, y ) m ( dy ) > m -a.e. x ∈ E then P is called pre-Harris .We can extend a stochastic operator P beyond the space L in the fol-lowing way. If 0 ≤ f n ≤ f n +1 , f n ∈ L , n ∈ N , then the pointwise almosteverywhere limit of f n exists and will be denoted by sup n f n . For f ≥ P f = sup n P f n for f = sup n f n , f n ∈ L . (Note that P f is independent of the particular approximating sequence f n and that P f may be infinite.) Moreover, if P is the transition operatorcorresponding to P then (9) holds for all measurable nonnegative f . A non-negative measurable f ∗ is said to be subinvariant (invariant) for a stochasticoperator P if P f ∗ ≤ f ∗ ( P f ∗ = f ∗ ).Let E ∗ ⊂ E be a given family of measurable subsets of E . A stochasticoperator P is called sweeping with respect to E ∗ iflim n →∞ Z B P n f ( x ) m ( dx ) = 0 for all f ∈ D ( m ) , B ∈ E ∗ . A nonnegative measurable f ∗ is said to be locally integrable with respect to E ∗ if Z B f ∗ ( x ) m ( dx ) < ∞ for all B ∈ E ∗ . Proposition 1 ([27, Corollary 3]) . Suppose that a stochastic operator P ispre-Harris and has no invariant density. If P has a subinvariant f ∗ with ELF-SIMILAR SOLUTIONS OF FRAGMENTATION EQUATIONS REVISITED 5 f ∗ > a.e. and f ∗ is locally integrable with respect to E ∗ , then the operator P is sweeping with respect to E ∗ . We conclude this section with the notion of stochastic semigroups andasymptotic behaviour of such semigroups. A family of stochastic operators { P ( t ) } t ≥ on L which is a C - semigroup , i.e.,(1) P (0) = I (the identity operator);(2) P ( t + s ) = P ( t ) P ( s ) for every s, t ≥ f ∈ L the mapping t P ( t ) f is continuous: for each s ≥ t → s + k P ( t ) f − P ( s ) f k = 0;is called a stochastic semigroup . A nonnegative measurable f ∗ is said tobe subinvariant (invariant) for the semigroup { P ( t ) } t ≥ if it is subinvariant(invariant) for each operator P ( t ).A stochastic semigroup { P ( t ) } t ≥ is called asymptotically stable if it hasan invariant density f ∗ such thatlim t →∞ k P ( t ) f − f ∗ k = 0 for all f ∈ D ( m )and partially integral if, for some s >
0, the operator P ( s ) is partially inte-gral. A stochastic semigroup is called sweeping with respect to E ∗ iflim t →∞ Z B P ( t ) f ( x ) m ( dx ) = 0 for all f ∈ D ( m ) , B ∈ E ∗ . Construction of Markov processes
In the first part of this section we construct two Markov processes suchthat (8) holds and their distributions are related to equations (1) and (4).The second part contains the proof of Theorem 1. At the end of this sectionwe discuss the case of α < ε n , θ n , n ∈ N , be sequences of independent random variables, wherethe ε n are exponentially distributed with mean 1 and the θ n are identicallydistributed with distribution function H on (0 ,
1) of the form H ( r ) = Pr( θ ≤ r ) = Z r h ( z ) zdz, r ∈ (0 , . (10)If Y is a positive random variable independent of θ n , n ∈ N , then thesequence Y n = θ n Y n − , n ≥ , defines a discrete-time Markov process with stochastic transition kernel P ( x, B ) = Z B ( zx ) h ( z ) zdz, x > , B ∈ B (0 , ∞ ) . (11)The transition operator P on L corresponding to P is as in (5).The process { Y ( t ) } t ≥ is a pure jump Markov process [28, Section 6.1]with the jump rate function a ( x ) = x α and the jump distribution P so thatthe process stays at x for a random time, which is called a holding timeand has an exponential distribution with mean 1 /a ( x ), and then it jumps WERONIKA BIEDRZYCKA AND MARTA TYRAN-KAMI ´NSKA according to the probability distribution P ( x, · ), independently on how longit stays at x . Therefore we define the sample path of { Y ( t ) } t ≥ starting at Y (0) = Y = x as Y ( t ) = Y n , τ n ≤ t < τ n +1 , n ≥ , (12)where τ n are the jump times τ = 0 , τ n := σ n + τ n − , n ≥ , and σ n are the holding times defined by σ n := ε n a ( Y n − ) = ε n Y αn − , n ≥ . Note that (see e.g. [28, Section 6.1]) if the probability density function of Y (0) satisfies Pr( Y (0) ∈ B ) = Z B c ( x ) x dx (13)where c ∈ D ( m ), thenPr( Y ( t ) ∈ B ) = Z B c ( t, x ) x dx for all t > , B ∈ B (0 , ∞ ) , (14)where c is the solution of equation (1) with initial condition c .The sample path of { X ( t ) } t ≥ starting from X (0) = X = x is defined as X ( t ) = e t − t n X n , t n ≤ t < t n +1 , n ≥ , where t n are the jump times t := 0 , t n = log (cid:18) ε n X αn − + 1 (cid:19) /α + t n − (15)and X n = X ( t n ) are the post-jump positions X n = θ n (cid:0) ε n + X αn − (cid:1) /α , n ≥ . (16)The process { X ( t ) } t ≥ , representing fragmentation with growth, is the min-imal piecewise deterministic Markov process [28, Section 6.2] with charac-teristics ( π, ϕ, P ), where ϕ ( x ) = αx α and π t x = e t x, x > , t ≥ . This is a particular example of a semiflow with jumps as studied in [11],where the jumps are defined by the mappings T θ ( x ) = θx and densities p θ ( x ) = h ( θ ) θ for x ∈ E = (0 , ∞ ), θ ∈ Θ = (0 , X (0) = Y (0) and Y (0) satisfies (13), thenPr( X ( t ) ∈ B ) = Z B P ( t ) c ( x ) xdx for all t > , B ∈ B (0 , ∞ ) , where { P ( t ) } t ≥ is a stochastic semigroup on L and u ( t, x ) = P ( t ) c ( x ) isthe solution of (4) with initial condition u (0 , x ) = c ( x ), see [28, Section 6.2]. ELF-SIMILAR SOLUTIONS OF FRAGMENTATION EQUATIONS REVISITED 7
This and (8) imply that the solution c of equation (1) with initial condition c can be represented as c ( t, x ) = γ ( t ) P (log γ ( t )) c ( γ ( t ) x ) , t > , x > , where γ ( t ) = (1 + t ) /α . (17)Hence, if the semigroup { P ( t ) } t ≥ is asymptotically stable with invariantdensity u ∗ thenlim t →∞ Z ∞ | c ( t, x ) − γ ( t ) u ∗ ( γ ( t ) x ) | x dx = lim t →∞ k P (log γ ( t )) c − u ∗ k = 0 . Consequently, this reduces the proof of Theorem 1 to the study of asymptoticstability of the semigroup { P ( t ) } t ≥ .For the proof of Theorem 1 we need the following result [11, Corollary3.16], which is a refinement of [11, Theorem 1.1]. Theorem 2.
Assume that the semigroup { P ( t ) } t ≥ is partially integral andthat the chain ( X ( t n )) n ≥ defined in (16) has only one invariant probabilitymeasure µ ∗ , absolutely continuous with respect to m . If the density f ∗ = dµ ∗ /dm is strictly positive a.e., then { P ( t ) } t ≥ is asymptotically stable ifand only if E ( t ) := Z ∞ E x ( t ) f ∗ ( x ) m ( dx ) < ∞ , (18) where t is the first jump time in (15) and E x denotes the expectation oper-ator with respect to the distribution P x of the process starting at X (0) = x . We first show that all assumptions of Theorem 2 are satisfied. We nextprove in Lemma 4 that conditions (18) and (7) are equivalent.
Lemma 1.
For each t > the operator P ( t ) is pre-Harris. In particular,the semigroup { P ( t ) } t ≥ is partially integral.Proof. Since the semigroup { P ( t ) } t ≥ is stochastic, we have m ( y : P y ( t ∞ < ∞ ) >
0) = 0 by [28, Corollary 5.3], where t ∞ = lim n →∞ t n . Observe that if y is such that P y ( t ∞ < ∞ ) = 0, then P y ( X ( t ) ∈ B ) = ∞ X n =0 P y ( X ( t ) ∈ B, t n ≤ t < t n +1 )and X ( t ) = e t − t n X ( t n ) for t ∈ [ t n , t n +1 ), n ≥
0. For n = 1 we have P y ( e t − t X ( t ) ∈ B, t ≤ t < t )= Z Z t B ( e t θy ) ψ t − s ( θe s y ) h ( θ ) θϕ ( e s y ) ψ s ( y ) dsdθ, where ψ t ( y ) = e − R t ϕ ( e r y ) dr . The change of variables x = e t θy leads to P y ( e t − t X ( t ) ∈ B, t ≤ t < t ) = Z B p ( x, y ) xdx, WERONIKA BIEDRZYCKA AND MARTA TYRAN-KAMI ´NSKA where p ( x, y ) = 1 (0 ,e t y ) ( x ) h (cid:18) xe t y (cid:19) e t y ) Z t ψ t − s ( e s − t x ) ϕ ( e s y ) ψ s ( y ) ds for x, y >
0. Hence Z B P ( t ) f ( x ) m ( dx ) = Z ∞ P y ( X ( t ) ∈ B ) f ( y ) m ( dy ) ≥ Z ∞ Z B p ( x, y ) m ( dx ) f ( y ) m ( dy )for all f ∈ D ( m ) and all Borel measurable sets B , which implies that P ( t ) f ( x ) ≥ Z ∞ p ( x, y ) f ( y ) m ( dy ) , f ∈ D ( m ) . Observe that Z ∞ p ( x, y ) m ( dy ) > m -a.e. x ∈ (0 , ∞ ) , which completes the proof. (cid:3) We will use the following lemma. Its proof is straightforward.
Lemma 2.
Assume that ξ and θ are independent random variables, where ξ has a probability density function f ξ on (0 , ∞ ) , while θ has a density f θ on (0 , . Then the density f ξθ of the random variable ξθ is given by f ξθ ( x ) = Z ∞ x f θ (cid:16) xr (cid:17) r f ξ ( r ) dr, x > , and it is positive a.e. if f ξ is positive a.e. Equality in distribution will be denoted by d =. Lemma 3.
Let ε n , θ n , n ∈ N , be sequences of independent random variables,where the ε n are exponentially distributed with mean and the θ n are iden-tically distributed with distribution function H as in (10) . Then the randomvariable X ∞ = X k ≥ ε k k Y j =1 θ αj /α (19) is finite a.e. and it satisfies X ∞ d = θ ( ε + X α ∞ ) /α (20) with independent X ∞ , θ , ε , where θ d = θ and ε d = ε . Moreover, E X α ∞ = E θ α − E θ α , (21) the distribution µ ∗ of X ∞ is absolutely continuous with respect to m withstrictly positive density f ∗ , and it is the unique stationary distribution of theMarkov chain ( X n ) n ≥ defined in (16) . ELF-SIMILAR SOLUTIONS OF FRAGMENTATION EQUATIONS REVISITED 9
Proof.
It follows from (16) that X αn = ε n θ αn + X αn − θ αn , n ≥ . By iterating this equation we obtain X αn = ε n θ αn + ε n − θ αn − θ αn + . . . + ε n Y j =1 θ αj + X α n Y j =1 θ αj . Since θ αj ∈ [0 , j ≥
1, we see that the sequence Q nj =1 θ αj , being monotone,converges almost surely. In fact, it converges to zero, by the strong law oflarge numbers. It is easily seen that X αn − X α n Y j =1 θ αj d = ξ n , n ≥ , where ξ n = ε θ α + ε θ α θ α + . . . + ε n n Y j =1 θ αj , and the sequence ξ n converges almost surely to X α ∞ , where X ∞ is as in (19).Therefore, if the Markov chain ( X n ) n ≥ has a stationary distribution thenit has to be the distribution of X ∞ .Let the random variable Z ∞ be defined by Z ∞ = X k ≥ ε k k − Y j =1 θ αj . (22)Note that in the right-hand side of (22), we take the product to be equalto 1 for k = 1. Since the random variables θ j , ε j are nonnegative, Z ∞ is awell defined random variable with values in [0 , ∞ ]; in fact, it is finite almostsurely [17, Theorem 2.1] for our choice of θ j , ε j . Since −∞ ≤ E (log θ ) < E (log(max { ε , } ) < ∞ , the series in (22) converges almost surely, by[29, Theorem 1.6], and Z ∞ d = θ α Z ∞ + ε , where θ , ε , Z ∞ are independent with θ d = θ , ε d = ε . The Markov chain( Z n ) n ≥ defined by Z n = θ αn Z n − + ε n , n ≥ , has a unique stationary distribution, by [29, Theorem 1.5], which is thedistribution of Z ∞ . Now, observe that θ α Z ∞ d = X α ∞ . Moreover, the random variable X α ∞ + ε has the same distribution as therandom variable Z ∞ . Consequently, the distribution of X ∞ is the uniquestationary distribution of the Markov chain ( X n ) n ≥ .Next observe that the distribution of Z ∞ , being a convolution of two dis-tributions one of which is absolutely continuous with respect to the Lebesgue measure, is absolutely continuous. Hence, it has a probability density func-tion f Z ∞ . Since X ∞ d = θ Z /α ∞ and the probability density function of Z /α ∞ is given by αx α − f Z ∞ ( x α ), the random variable X ∞ also has a probabilitydensity function f X ∞ , which implies that f ∗ ( x ) x = f X ∞ ( x ) for x >
0. Toshow that f ∗ is positive a.s. it is enough to show, by Lemma 2, that f Z ∞ ispositive a.e. Since Z ∞ d = X α ∞ + ε , we have f Z ∞ ( x ) = Z x e − ( x − y ) f X α ∞ ( y ) dy which shows that f Z ∞ is positive on an interval ( x , ∞ ), where x ≥
0. Tocomplete the proof, it remains to show that x = 0. From (22) it followsthat f Z ∞ ( x ) satisfies the following equation f Z ∞ ( x ) = Z x Z ∞ z g α (cid:16) zy (cid:17) y f Z ∞ ( y ) dye − ( x − z ) dz, where g α is the probability density function of the random variable θ α . Bychanging the order of integration, we obtain f Z ∞ ( x ) = Z ∞ Z min { x,y } g α (cid:16) zy (cid:17) y e − ( x − z ) dzf Z ∞ ( y ) dy. Suppose that x >
0. Then for every x < x and every y > x we obtain Z x g α (cid:16) zy (cid:17) y e − ( x − z ) dz = 0This implies that for every r < R r g α ( t ) dt = 0 , which contradictsthe fact that R g α ( t ) dt = 1 and shows that x = 0. Consequently, thedensity f ∗ is positive a.e.Finally, observe that we have 0 ≤ E θ α <
1. In fact, since θ α ∈ [0 , E θ α ∈ [0 , E θ α = 1. Then θ α = 1 a.e., which impliesthat θ = 1 a.e. Hence θ has no density, which leads to a contradiction.From (19) we calculate E X α ∞ = X k ≥ E ε k k Y j =1 E θ αj = X k ≥ k Y j =1 E θ α = X k ≥ ( E θ α ) k = E θ α − E θ α , which gives (21) and completes the proof. (cid:3) Lemmas 1 and 3 imply that all assumptions of Theorem 2 hold. Theorem 1now follows by combining Theorem 2 and the next lemma.
Lemma 4.
The random variable t in (15) satisfies (18) if and only if E log θ > −∞ , in which case E ( t ) = E ( − log θ ) = − Z log z h ( z ) zdz. ELF-SIMILAR SOLUTIONS OF FRAGMENTATION EQUATIONS REVISITED 11
Proof.
We have t = α log ε + X α X α , where ε and X are independent. Thisleads to 1 α log ε − log X ≤ t ≤ α ( ε + X α − − log X . To calculate the first moment of t we take X d = X ∞ . Since X α d = X α ∞ isintegrable by (21) and E log ε = − γ , where γ is the Euler–Mascheroni con-stant, we obtain that t has a finite first moment if and only if | E log X ∞ | < ∞ . We have E ( t ) = Z ∞ Z ∞ log (cid:16) yx α + 1 (cid:17) α e − y dyf ∗ ( x ) m ( dx )= Z Z ∞ Z ∞ (cid:16) log θ ( y + x α ) α − log x − log θ (cid:17) e − y dyf ∗ ( x ) m ( dx ) dθ = E log X ∞ − E log X ∞ − E log θ = − E log θ . Moreover, from (20) it follows thatlog X α ∞ d = α log θ + log ε + log (cid:18) X α ∞ ε (cid:19) ≥ α log θ + log ε . On the other hand,log X α ∞ d = α log θ + log( ε + X α ∞ ) ≤ α log θ + ε + X α ∞ − . Hence E log θ + 1 α E log ε ≤ E log X ∞ ≤ E log θ + 1 α E X α ∞ , which implies that E log θ > −∞ if and only if | E log X ∞ | < ∞ and com-pletes the proof. (cid:3) We conclude this section with a construction of the jump Markov process { Y ( t ) } t ≥ corresponding to the fragmentation equation (1) with a ( x ) = x α and α <
0. The sample path of Y ( t ) starting at Y (0) = Y is defined as in(12) as long as t ∈ [ τ n , τ n +1 ) for some n . Since Y α τ n = Y α n X k =1 ε k Y αk − = n X k =1 ε k k − Y j =1 θ − αj , n ≥ , and − α >
0, we see, as in the proof of Lemma 3, that the limitlim n →∞ Y α τ n = ∞ X k =1 ε k k − Y j =1 θ − αj =: I ∞ exists and is finite a.s. Thus the explosion time of the process, being definedby τ ∞ = lim n →∞ τ n , is finite a.s. The sequence Y n , n ≥
0, is non-increasingand converging to 0 a.s. Consequently, we can set Y ( t ) = 0 for t ≥ τ ∞ andsay that Y − α I ∞ is the first time when Y ( t ) reaches 0. Observe that we have Y ( t ) = (1 + t ) − /α X (log(1 + t ) − /α ) , t ≥ , where the corresponding piecewise deterministic Markov process { X ( t ) } t ≥ has characteristics ( π, ϕ, P ) with ϕ ( x ) = | α | x α , π t x = e − t x, x > , t ≥ , and the evolution equation as in [28, Section 6.3] or [1, Section 4].Another representation of { Y ( t ) } t ≥ is as follows. Let { Z ( t ) } t ≥ be acompound Poisson process of the form Z ( t ) = − N ( t ) X j =1 log θ j , t > , where { N ( t ) } t ≥ is a Poisson process with jump times ˜ τ n = P nj =1 ε j , n ≥ { Y ( t ) } t ≥ can be represented in the form [19] Y ( t ) = Y e − Z ( ρ ( Y α t )) , t ≥ , where ρ is the time-change given by ρ ( t ) = inf { r ≥ Z r e αZ ( s ) ds > t } , t ≥ . Note that Z ρ ( t )0 e αZ ( s ) ds = t if and only if t < I ∞ , and ρ ( t ) = + ∞ otherwise. The random variable I ∞ isan example of the so-called exponential functional (see e.g. [13]) I ∞ = Z ∞ e αZ ( s ) ds. This can be easily seen by noting that N ( s ) = k for t ∈ [˜ τ k , ˜ τ k +1 ) with˜ τ := 0, ε k +1 = ˜ τ k +1 − ˜ τ k , k ≥
0, and Z ∞ e αZ ( s ) ds = X k ≥ Z ˜ τ k +1 ˜ τ k k Y j =1 θ − αj ds = ˜ τ + X k ≥ (˜ τ k +1 − ˜ τ k ) k Y j =1 θ − αj = I ∞ . To get finiteness of I ∞ in terms of pathwise properties of the process { Z ( t ) } t ≥ ,one can simply assume that E ( Z (1)) ∈ (0 , ∞ ), which is equivalent to (7).4. Examples and final remarks
We have proved that the homogeneous fragmentation equation has a self-similar solution if and only if E (log θ ) > −∞ where θ is a random variablewith distribution function H as in (10). In that case the growth fragmenta-tion equation has an integrable stationary solution. Here we give the formulafor the stationary solution in terms of the probability density function of therandom variable Z ∞ as defined in (22).Using the notation of [11] note that the stochastic transition kernel K ofthe Markov chain ( X n ) n ≥ defined in (16) is of the form K ( x, B ) = Z ∞ P ( e s x, B ) ϕ ( e s x ) e − R s ϕ ( e r x ) dr ds, x > , B ∈ B (0 , ∞ ) , ELF-SIMILAR SOLUTIONS OF FRAGMENTATION EQUATIONS REVISITED 13 see e.g. [11, Theorem 3.14], where ϕ ( x ) = αx α and P is the stochastictransition kernel defined in (11). We have Z ∞ K ( x, B ) f ( x ) m ( dx ) = Z ∞ P ( x, B ) ϕ ( x ) R f ( x ) m ( dx ) (23)for all B ∈ B (0 , ∞ ) and f ∈ D ( m ), where R f ( x ) = Z ∞ e − s e ( e − s x ) α − x α f ( e − s x ) ds, x > , f ∈ D ( m ) . Since P in (5) is the transition operator corresponding to P , we obtain, by(23) and (9), Z ∞ K ( x, B ) f ( x ) m ( dx ) = Z B P ( ϕR f )( x ) m ( dx ) , B ∈ B (0 , ∞ ) , f ∈ D ( m ) . Hence the stochastic operator K , being the transition operator on L corre-sponding to K , is given by Kf = P ( ϕR f ) , f ∈ D ( m ) . Note that f ∗ in Lemma 3 is the unique invariant density of the stochasticoperator K , thus f ∗ = P ( ϕf ∗ ) , where f ∗ = R f ∗ . It follows from [11, Theorem 3.3, Proposition 3.8] that f ∗ is subinvariantfor the semigroup { P ( t ) } t ≥ . We have f ∗ > { P ( t ) } t ≥ can have at most one invariantdensity, by [11, Theorem 3.15]. Note that f ∗ might not be integrable, buttaking B = (0 , ∞ ) and f = f ∗ in (23) shows that Z ∞ ϕ ( x ) f ∗ ( x ) m ( dx ) = Z ∞ f ∗ ( x ) m ( dx ) = 1 . (24)From the proof of [11, Theorem 3.15] it follows that E ( t ) = k f ∗ k . Thus k f ∗ k = E ( − log θ ), by Lemma 4.On the other hand, f ∗ ( x ) x is the probability density function of the ran-dom variable X ∞ d = θ Z /α ∞ with θ d = θ and the operator P correspondsto a multiplication by θ . Thus the probability density function of Z /α ∞ satisfies αx α − f Z ∞ ( x α ) = ϕ ( x ) f ∗ ( x ) x. Consequently, if E ( − log θ ) < ∞ then the semigroup { P ( t ) } t ≥ has a uniqueinvariant density u ∗ and it is given by u ∗ ( x ) = f ∗ ( x ) k f ∗ k = f Z ∞ ( x α ) E ( − log θ ) x . We have proved the following.
Proposition 2. If (7) holds, equivalently E ( − log θ ) < ∞ , then the self-similar solution of equation (1) in Theorem 1 is of the form c ∗ ( t, x ) = f Z ∞ ((1 + t ) x α ) E ( − log θ ) x , x, t > , where f Z ∞ is the probability density function of the random variable Z ∞ in (22) . This form of the self-similar solution should be compared with the scalingassumption in [32, equation (15)], where the function Φ corresponds to f Z ∞ .We now give an exactly solvable example, known since the [26]. Example . Consider the function h ( z ) = βz β − with β >
0. Then θ d = U /β , where U is a random variable with uniform distribution on [0 , Z ∞ has the gamma distribution with shape parameter 1 + β/α (see e.g. [29,Example 3.8]), we have f Z ∞ ( x ) = 1Γ(1 + β/α ) x β/α e − x , x > , where Γ is the gamma function. This implies that u ∗ ( x ) = β Γ(1 + β/α ) x β − e − x α = α Γ( β/α ) x β − e − x α . Note that u ∗ ( x ) x is the probability density function of the generalized gammadistribution with parameters ( α, β, Z ∞ can be represented as the exponential functional Z ∞ d = Z ∞ e − αZ ( t ) dt of the compound Poisson process { Z ( t ) } t ≥ defined above. The Laplaceexponent φ ( q ) of { Z ( t ) } t ≥ , which is defined by E ( e − qZ ( t ) ) = e − tφ ( q ) , t > , is of the form φ ( q ) = E (1 − θ q ) = Z (1 − z q ) zh ( z ) dz, q > . Hence, [13, Proposition 3.3] implies that the random variable Z ∞ is deter-mined by its moments E ( Z n ∞ ) = n ! Q nk =1 φ ( αk ) = n ! Q nk =1 E (1 − θ αk ) , n = 1 , , . . . . We next give two examples where the random variable Z ∞ can be identi-fied through its moments. Example . Recall that a random variable θ has a beta distribution withparameters ( a, b ), a, b >
0, if its probability density function is f θ ( x ) = Γ( a + b )Γ( a )Γ( b ) x a − (1 − x ) b − , x ∈ (0 , . ELF-SIMILAR SOLUTIONS OF FRAGMENTATION EQUATIONS REVISITED 15 If θ is a product of two independent random variables with beta distribu-tions with parameters ( β ,
1) and ( β , E ( − log θ ) = 1 /β + 1 /β and Z ∞ is a product of two independent random variables, one is beta dis-tributed with parameters (1+ a , a ) and the other has a gamma distributionwith shape parameter 1 + a , where a = β α , a = β α . (25) Remark . It is easily seen that if Z ∞ = θξ where θ and ξ are independentrandom variables, θ has a beta distribution with parameters (1 + a , a ) and ξ has a gamma distribution with shape parameter 1 + a , then the probabilitydensity function of Z ∞ is of the form f Z ∞ ( x ) = Γ(1 + a + a )Γ(1 + a )Γ(1 + a ) e − x x a U ( a, a − a , x ) , x > , where U is the confluent hypergeometric function of the second kind U ( a, b, x ) = 1Γ( a ) Z ∞ e − xs s a − (1 + s ) b − a − ds, a, x > , b ∈ R . Example . As in [32] consider now the function h ( z ) = pβ z β − + (1 − p ) β z β − , z ∈ (0 , , where β , β > p ∈ [0 , β > β and p >
0. Then p ( a − a ) >
0, where a , a are as in (25), E ( − log θ ) = p/β + (1 − p ) /β ,and Z ∞ is the product of two independent random variables, one is betadistributed with parameters (1 + a , p ( a − a )) and the other has a gammadistribution with parameter 1 + a . Remark 1 and Proposition 2 allow us torecover the scaling solutions from [32].We conclude the paper by commenting on the behaviour of solutions ofthe fragmentation equation when E (log θ ) = −∞ . Proposition 3. If (7) does not hold then, for every solution c of (1) withinitial condition c ∈ D ( m ) , we have lim t →∞ Z ∞ x /γ ( t ) c ( t, x ) x dx = 0 for all x > . Proof. If E ( − log θ ) = ∞ then f ∗ is not integrable and the semigroup { P ( t ) } t ≥ has no invariant density, since if there were one, then it wouldbe a scalar multiple of f ∗ , by [11, Corollaries 3.11, 3.12]. Hence, for every s > P ( s ) does not have an invariant density, by [20, Proposi-tion 7.12.1]. From Lemma 1 and Proposition 1 it follows that the operator P ( s ) is sweeping which together with [20, Theorem 7.11.1] implies that thesemigroup { P ( t ) } t ≥ is sweeping from every set B satisfying Z B f ∗ ( x ) m ( dx ) < ∞ . We have ϕ ( x ) = αx α ≥ ϕ ( x ) > x ≥ x >
0. From this and (24) wesee that f ∗ is integrable over intervals B = ( x , ∞ ), x >
0. Consequently,lim t →∞ Z ∞ x P ( t ) c ( x ) xdx = 0 , which together with (17) completes the proof. (cid:3) Remark . It is interesting to note that every solution c of (1) with initialcondition c ∈ D ( m ) satisfieslim t →∞ Z ∞ x c ( t, x ) xdx = 0 for all x > . To see this observe that if (7) does not hold then this is a consequence ofProposition 3, since γ ( t ) ≥ c ≥
0, while if (7) holds then Z ∞ x c ( t, x ) xdx ≤ Z ∞ | c ( t, x ) − c ∗ ( t, x ) | xdx + Z ∞ x c ∗ ( t, x ) xdx and the right-hand side converges to zero as t → ∞ , by Theorem 1 andintegrability of u ∗ . Acknowledgments
We are indebted to the anonymous referees for their comments that havematerially improved this paper.
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