Fractional pure birth processes

Transcription

1 Bernoulli 6(3), 2, DOI:.35/9-BEJ235 arxiv:8.245v2 [math.pr] 4 Feb 2 Fractional pure birth processes ENZO ORSINGHER * and FEDERICO POLITO ** Dipartimento di Statistica, Probabilità e Stat. Appl., Sapienza Università di Roma, pl.a. Moro 5, 85 Rome, Italy. s: * enzo.orsingher@uniroma.it; ** federico.polito@uniroma.it We consider a fractional version of the classical nonlinear birth process of which the Yule Furry model is a particular case. Fractionality is obtained by replacing the first order time derivative in the difference-differential equations which govern the probability law of the process with the Dzherbashyan Caputo fractional derivative. We derive the probability distribution of the number N ν(t) of individuals at an arbitrary time t. We also present an interesting representation for the number of individuals at time t, in the form of the subordination relation N ν(t) N(T 2ν(t)), where N(t) is the classical generalized birth process and T 2ν(t) is a random time whose distribution is related to the fractional diffusion equation. The fractional linear birth process is examined in detail in Section 3 and various forms of its distribution are given and discussed. Keywords: Airy functions; branching processes; Dzherbashyan Caputo fractional derivative; iterated Brownian motion; Mittag Leffler functions; nonlinear birth process; stable processes; Vandermonde determinants; Yule Furry process. Introduction We consider a birth process and denote by N(t), t>, the number of components in a stochastically developing population at time t. Possible examples are the number of particles produced in a radioactive disintegration and the number of particles in a cosmic ray shower where death is not permitted. The probabilities p (t)pr{n(t)} satisfy the difference-differential equations where, at time t, dp dt p + p,, (.) {,, p (), 2. (.2) This means that we initially have one progenitor igniting the branching process. For information on this process, consult Gihman and Sorohod [5], page 322. This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2, Vol. 6, No. 3, This reprint differs from the original in pagination and typographic detail c 2 ISI/BS

2 Fractional birth processes 859 Here, we will examine a fractional version of the birth process where the probabilities are governed by d ν p dt ν p + p,, (.3) and where the fractional derivative is understood in the Dzherbashyan Caputo sense, that is, as d ν p dt ν t (d/ds)p (s) Γ( ν) (t s) ν ds for <ν < (.4) (see Podlubny [2]). The use of a Dzherbashyan Caputo derivative is preferred because in this case, initial conditions can be expressed in terms of integer-order derivatives. Extensions of continuous-time point processes lie the homogeneous Poisson process to the fractional case have been considered in Jumarie [7], Cahoy [3], Lasin [9], Wang and Wen [7], Wang, Wen and Zhang [8], Wang, Zhang and Fan [9], Uchaiin and Sibatov [5], Repin and Saichev [3] and Beghin and Orsingher [2]. A recently published paper(uchaiin, Cahoy and Sibatov[6]) considers a fractional version of the Yule Furry process where the mean value EN ν (t) is analyzed. By recursivelysolvingequation (.3) (we write p (t), t>, in equations(.3) and p ν (t) for the solutions), we obtain that p ν (t) Pr{N ν (t)} { } j j m l,l m ( l m ) E ν,( m t ν ), >, E ν, ( t ν ),. (.5) Result (.5) generalizes the classical distribution of the birth process (see Gihman and Sorohod [5], page 322, or Bartlett [], page 59), where, instead of the exponentials, we have the Mittag Leffler functions, defined as E ν, (x) h x h, x R,ν >. (.6) Γ(νh+) The fractional pure birth process has some specific features entailed by the fractional derivative appearing in (.4), which is a non-local operator. The process governed by fractionalequations(and therefore the related probabilities p ν (t)pr{n ν(t)}, ) displays a slowly decreasing memory which seems a characteristic feature of all real systems (for example, the hereditariety and the related aspects observed in phenomena such as metal fatigue, magnetic hysteresis and others). Fractional equations of various types have proven to be useful in representing different phenomena in optics (light propagation through random media), transport of charge carriers and also in economics (a survey of applications can be found in Podlubny [2]). Below, we show that for the linear birth process N ν (t),t>, the mean values EN ν (t), VarN ν (t) are increasing functions as the

3 86 E. Orsingher and F. Polito order of fractionality ν decreases. This shows that the fractional birth process is capable of representing explosively developing epidemics, accelerated cosmic showers and, in general, very rapidly expanding populations. This is a feature which the fractional pure birth process shares with its Poisson fractional counterpart whose practical applications have been studied in recent wors (see, for example, Lasin [9] and Cahoy [3]). We are able to show that the fractional birth process N ν (t) can be represented as N ν (t)n(t 2ν (t)), t>,<ν, (.7) where T 2ν (t), t>, is the random time process whose distribution at time t is obtained from the fundamental solution to the fractional diffusion equation (the fractional derivative is defined in (.4)) 2ν u t 2ν 2 u s2, <ν, (.8) subject to the initial conditions u(s,) δ(s) for < ν and also u t (s,) for /2<ν, as { Pr{T 2ν (t) ds} 2u2ν (s,t)ds for s>, (.9) for s<. This means that the fractional birth process is a classical birth process with a random time T 2ν (t) which is the sole component of (.7) affected by the fractional derivative. In equation (.8) and throughout the whole paper, the fractional derivative must be understood in the Dzherbashyan Caputo sense (.3). The representation (.7) leads to where Pr{N ν (t)} e ms j Pr{N(s)} j m l,l m ( l m ), e s, Pr{N(s)}Pr{T 2ν (t) ds}, (.) >, s>,, s>. (.) Formula (.) immediately shows that Pr{N ν(t) } if and only if Pr{N(t) }. It is well nown that the process N(t), t >, is such that Pr(N(t) < ) for all t > (non-exploding) if (see Feller [4], page 452). A special case of the above fractional birth process is the fractional linear birth process where. In this case, the distribution (.5) reduces to the simple form p ν (t) j ( ) j E j ν, ( jt ν ),,t>. (.2)

4 Fractional birth processes 86 For ν, we retrieve from (.2) the classical geometric structure of the linear birth process with a single progenitor, that is, p (t)( e t ) e t,,t>. (.3) An interesting qualitative feature of the fractional linear birth process can be extracted from (.2); it permits us to highlight the dependence of the branching speed on the order of fractionality ν. We show in Section 3 that Pr{N ν (dt)n + N ν ()n } n (dt) ν Γ(ν +) (.4) and this proves that a decrease in the order of fractionality ν speeds up the reproduction of individuals. We are not able to generalize (.4) to the case Pr{N ν (t+dt)n + N ν (t)n } (.5) because the process we are investigating is not time-homogeneous. For the fractional linear birth process, the representation (.7) reduces to the form N ν (t)n(t 2ν (t)), t>,<ν, (.6) and has an interesting special structure when ν /2 n. For example, for n 2, the random time appearing in (.6) becomes a folded iterated Brownian motion. This means that N /4 (t)n( B ( B 2 (t) ) ). (.7) Clearly, B 2 (t) is a reflecting Brownian motion starting from zero and B ( B 2 (t) ) is a reflecting iterated Brownian motion. This permits us to write the distribution of (.7) in the following form: Pr{N /4 (t) N /4 ()} ( e s ) e s { 2 2 } e s2 /(4ω) e ω2 /(4t) dω ds. 2π2ω 2π2t The case ν /2 n involves the (n )-times iterated Brownian motion (.8) I n (t)b ( B 2 ( B n (t) ) ) (.9) with distribution Pr{ B ( B 2 ( B n (t) )) ds} ds2 n e s2 /(4ω ) 4πω dω For details on this point, see Orsingher and Beghin []. e ω2 /(4ω2) e ω2 n /(4t) dω 2 dω n. 4πω2 4πt (.2)

5 862 E. Orsingher and F. Polito 2. The distribution function for the generalized fractional birth process We now present the explicit distribution Pr{N ν (t) N ν ()}p ν (t), t>,,<ν, (2.) of the number of individuals in the population expanding according to (.3). Our technique is based on successive applications of the Laplace transform. Our first result is the following theorem. Theorem 2.. The solution to the fractional equations d ν p dt ν p + p,,<ν, {,, p (), 2, (2.2) is given by p ν (t) Pr{N ν(t) N ν ()} { } j j m l,l m ( l m ) E ν,( m t ν ), >, E ν, ( t ν ),. (2.3) Proof. We prove the result (2.3) by a recursive procedure. For, the equation is immediately solved by d ν p dt ν p, p (), (2.4) p ν (t)e ν,( t ν ). (2.5) For 2, equation (.3) becomes d ν p 2 dt ν 2p 2 + E ν, ( t ν ), p 2 (). (2.6) In view of the fact that e µt E ν, ( t ν )dt µν µ ν +, (2.7)

6 Fractional birth processes 863 the Laplace transform of (2.6) yields L 2 (µ) µ ν 2 [ µ ν + µ ν + 2 In the light of (2.7), from (2.8), we can determine the probability p ν 2 (t): Now, the Laplace transform of yields, after some computation, From this result, it is clear that ]. (2.8) p ν 2 (t)[e ν,( t ν ) E ν, ( 2 t ν )]. (2.9) 2 d ν p 3 dt ν 3p [E ν, ( t ν ) E ν, ( 2 t ν )] (2.) [ L 3 (µ) 2 µ ν ( 2 )( 3 ) µ ν + + ( 2 )( 3 2 ) µ ν (2.) + 2 ] + ( 3 )( 2 3 ) µ ν. + 3 [ p ν 3 (t) 2 ( 2 )( 3 ) E ν,( t ν ) + + ( 2 )( 3 2 ) E ν,( 2 t ν ) (2.2) ( 3 )( 2 3 ) E ν,( 3 t ν ) The procedure for >3 becomes more complicated. However, the special case 4 is instructive and so we treat it first. The Laplace transform of the equation d ν p 4 dt ν 4p [ ]. ( 2 )( 3 ) E ν,( t ν ) + + ( 2 )( 3 2 ) E ν,( 2 t ν ) (2.3) ( 3 )( 2 3 ) E ν,( 2 t ν ) ],

7 864 E. Orsingher and F. Polito subject to the initial condition p 4 (), becomes [ { } L 4 (µ) 2 3 µ ν ( 2 )( 3 )( 4 ) µ ν + µ ν + 4 { } + ( 2 )( 3 2 )( 4 2 ) µ ν + 2 µ ν + 4 { }] + ( 3 )( 2 3 )( 4 3 ) µ ν + 3 µ ν. + 4 (2.4) The critical point of the proof is to show that (( 3 2 )( 4 2 )( 4 3 ) ( 3 )( 4 )( 4 3 ) +( 2 )( 4 )( 4 2 )) ( 2 )( 3 )( 4 )( 3 2 )( 4 2 )( 4 3 ) ( 4 )( 2 4 )( 3 4 ). (2.5) We note that det det det 3 4 +det 2 4 det 2 3 (2.6) ( 3 2 )( 4 2 )( 4 3 ) ( 3 )( 4 )( 4 3 ) +( 2 )( 4 )( 4 2 ) ( 2 )( 3 )( 3 2 ), where, in the last step, the Vandermonde formula is applied. By inserting (2.6) into (2.4), we now have that [ L 4 (µ) 2 3 µ ν ( 2 )( 3 )( 4 ) µ ν + + ( 2 )( 3 2 )( 4 2 ) µ ν + 2

8 Fractional birth processes ( 3 )( 2 3 )( 4 3 ) µ ν + 3 ] + ( 4 )( 2 4 )( 3 4 ) µ ν + 4 (2.7) so that by inverting (2.7), we obtain the following result: p ν 4 (t) 3 j j{ 4 m } 4 l,l m ( l m ) E ν,( m t ν ). (2.8) We now tacle the problem of showing that (2.3) solves the Cauchy problem (2.2) for all >, by induction. This means that we must solve d ν p dt ν p + p (), j { j m l,l m ( l m ) E ν,( m t ν ) }, >4. (2.9) The Laplace transform of (2.9) reads We must now prove that [ µ ν L (µ) j j m l,l m ( l m ) µ ν + m ] (2.2) µν µ ν + l,l m (. l m ) m m l,l m ( l m ) l,l ( l ) (2.2) and this relation is also important for the proof of (.). In order to prove (2.2), we rewrite the left-hand side as m h l>h ( l h ) l,l m ( l m ) h l>h ( l h ) (2.22)

9 866 E. Orsingher and F. Polito and concentrate our attention on the numerator of (2.22). By analogy with the calculations in (2.6), we have that det 2 m 2 2 m ( ) m det m m h l>h ( l h ) l,l m ( l m ) m m+ 2 2 m 2 m+ 2 m h l>h ( l h ) l,l m ( l m ) + (2.23) h l>h ( l h ) l,l ( l ). In the third step of (2.23), we applied the Vandermonde formula and considered the fact that the nth column is missing. It must also be taen into account that l> ( l ) l>2 ( l 2 ) l>m ( l m ) ( m ) ( m 2 ) ( m m ) l>m ( l m ) l>m ( l m ) ( l m+ ) ( l ) (2.24) l>m+ h l>h ( l h ) ( ) m l,l m ( l m ). From (2.22) and (2.23), we have that m l,l m ( l m ) m h l>h ( l h ) l,l m ( l m ) l,l ( l ). l> h l>h ( l h ) (2.25) In view of (2.25), we can write that L (µ) j j m µ ν l,l m ( l m ) µ ν (2.26) + m because the th term of (2.26) coincides with the last term of (2.2) and therefore, by inversion of the Laplace transform, we get (2.3).

10 Fractional birth processes 867 Remar 2.. We now prove that for the generalized fractional birth process, the representation N ν (t)n(t 2ν (t)), t>,<ν, (2.27) holds. This means that the process under investigation can be viewed as a generalized birth process at a random time T 2ν (t), t>, whose distribution is the folded solution to the fractional diffusion equation (.8). where e µt G ν (u,t)dt by (2.3) 2 u j { 2 m } E ν, ( m t ν ) j j m ( j m ) +ue ν,( t ν ) e µt dt u j m µ ν j µ ν + m j m ( j m ) + uµν µ ν (2.28) + { 2 } µ ν + j m) j m ( +ue j m ) e s(µν s(µν + ) ds u j m G(u,s)µ ν e sµν ds G(u, s) { } e µt G(u,s)f T2ν (s,t)ds dt, e µt f T2ν (s,t)dtds e µt f T2ν (s,t)dtµ ν e sµν, s>, (2.29) is the Laplace transform of the folded solution to (.8). From (2.28), we infer that G ν (u,t) and from this, the representation (2.27) follows. G(u,s)f T2ν (s,t)ds (2.3) Remar 2.2. The relation (2.27) permits us to conclude that the functions (2.3) are non-negative because Pr{N ν (t)} Pr{N(s)}Pr{T 2ν (t) ds}, (2.3) and Pr{N(s) } > and Pr{N(s) }, as shown, for example, in Feller [4], page 452. Furthermore, the fractional birth process is non-exploding if and only if (/ ) for all values of <ν.

11 868 E. Orsingher and F. Polito 3. The fractional linear birth process In this section, we examine in detail a special case of the previous fractional birth process, namely the fractional linear birth process which generalizes the classical Yule Furry model. The birth rates in this case have the form, >,, (3.) and indicate that new births occur with a probability proportional to the size of the population. We denote by N ν (t) the number of individuals in the population expanding according to the rates (3.) and we have that the probabilities p ν (t)pr{n ν (t) N ν ()},, (3.2) satisfy the difference-differential equations d ν p dt ν p +()p, {,, p (), 2. <ν,, (3.3) The distribution (3.2) can be obtained as a particular case of (2.3) or directly, by means of a completely different approach, as follows. Theorem 3.. The distribution of the fractional linear birth process with a simple initial progenitor has the form p ν (t) Pr{N ν(t) N ν ()} j ( ) j E j ν, ( jt ν ),,<ν, (3.4) where E ν, (x) is the Mittag Leffler function (.6). Proof. We can prove the result (3.4) by solving equation (3.3) recursively. This means that p ν (t) hastheform(3.4),sopν (t) maintainsthesamestructure.thisistantamount to solving the Cauchy problem d ν p (t) dt ν p (), p (t)+() >. j j 2 ( ) j E j ν, ( jt ν ), (3.5) By applying the Laplace transform L,ν (µ) e µt p (t)dt to (3.5), we have that { ( } 2 j µ L,ν (µ)() )( ) j µ ν +j µ ν +. (3.6)

12 Fractional birth processes 869 Conveniently, the Laplace transform (3.6) can be written as {[ L,ν (µ)µ ν µ ν + µ ν + + ()( 2) [ 2 [ +()( ) 2 µ ν j This permits us to conclude that ] [ () µ ν +3 µ ν + µ ν +2 ] + µ ν + ]} µ ν +() µ ν + ( ) j j µ ν +j µν µ ν + L,ν (µ)µ ν j j ( j ] ) ( ) j. (3.7) ( ) j j µ ν +j. (3.8) By inverting (3.8), we immediately arrive at the result (3.4). For ν, (3.8) can be written as e µt p (t)dt e t e µt ( ) j e jt dt j j e µt {e t ( e t ) }dt (3.9) and this is an alternative derivation of the Yule Furry linear birth process distribution. Remar 3.. An alternative form of the distribution (3.4) can be derived by explicitly writing the Mittag Leffler function and conveniently manipulating the double sums obtained. We therefore have p ν (t) ( t ν ) m Γ(νm+) m + m j ( t ν ) m Γ(νm+) ( ) j (j +) m j j (tν ) ()! Γ(ν()+) + m ( ) j (j +) m (3.) j ( t ν ) m Γ(νm+) j ( ) j (j +) m. j

13 87 E. Orsingher and F. Polito The last step of (3.) is justified by the following formulas (see.54(6) and.54(5) on page 4 of Gradshteyn and Ryzhi [6]): N ( ( ) N ) (α+) n, valid for N n, (3.) n ( ) n (α+) n ( ) n n!. (3.2) What is remarable about (3.2) is that the result is independent of α. This can be ascertained as follows: S α n n ( ) n n n α r n r r r n r n n α r r ( ) ( n ) n r+. (3.3) By formula.54(3) on page 4 of Gradshteyn and Ryzhi [6], the inner sum in the third member of (3.3) equals zero for n r+ n (that is, for r n). Therefore (see formula.54(4) on page 4 of Gradshteyn and Ryzhi [6]), S α n n n α ( ) ( n ) n ( ) n n!. (3.4) We now provide a direct proof that the distribution (3.4) sums to unity. This is based on combinatorial arguments and will subsequently be validated by resorting to the representation of N ν (t) as a composition of the Yule Furry model with the random time T 2ν (t). Theorem 3.2. The distribution (3.4) is such that p ν (t) j ( ) j E j ν, ( jt ν ). (3.5) Proof. We start by evaluating the Laplace transform L ν (µ) of (3.5) as follows: L ν (µ) j µν ( j µν )( ) j µ ν +j j ( ) j j µ ν /++j. (3.6)

14 Fractional birth processes 87 A crucial role is played here by the well-nown formula (see Kirschenhofer [8]) Therefore, N N ( ) x+ N! x(x+) (x+n). (3.7) L ν (µ) µν µν µν µν l l ()! (µ ν /+)(µ ν /+2) (µ ν /+) Γ(l+)Γ(µ ν /+) Γ(µ ν /++(l+)) B (l+, ) µν + µν ( x) µν / dx e µt dt, x l ( x) µν / dx l (3.8) where B(p,q) xp ( x) q dx for p,q>. This concludes the proof of (3.5). The presence of alternating sums in (3.4) imposes the chec that p ν (t) for all. This is the purpose of the next remar. Remar 3.2. In order to chec the non-negativity of (3.4), we exploit the results of the proof of Theorem 3.2, suitably adapted. The expression e µt p ν µν (t)dt which emerges from (3.8) permits us to write ) B (, µν + (3.9) e µt p ν (t)dt x µν ( x)µν / dx x µν ν e(µ /)ln( x) dx x µν ν e µ / r xr /r dx x µν ν e µ x/ e µν x r /(r) dx. r2 (3.2)

15 872 E. Orsingher and F. Polito The terms e µν x r /(r) Ee µxr e µt qν r (x,t)dt (3.2) are the Laplace transforms of stable random variables X r S(σ r,,), where σ r ( xr πν r cos 2 )/ν (for details on this point, see Samorodnitsy and Taqqu [4], page 5). The term µν x 2 exp( µν ) is the Laplace transform of the solution of the fractional diffusion equation 2ν u u t 2ν 2 2 x2, <ν, (3.22) u(x,)δ(x), with the additional condition that u t (x,) for /2<ν, and can be written as u 2ν (x,t) t p ν (x,s) ds (3.23) 2Γ( ν) (t s) ν (see formula (3.5) in Orsingher and Beghin []), where p ν (x,)qν(x,) is the stable law with σ ( x πν cos 2 )/ν. We can represent the product µ ν e xµν / e µν x r /(r) r2 e µt { t } u 2ν (x,s)q ν (x,t s)ds dt, (3.24) where e µt q ν (x,t)dt r2 e µν x r /(r). (3.25) Thus q ν (x,t) appears as an infinite convolution of stable laws whose parameters depend on r and x. In the light of (3.24), we therefore have that e µt p ν (t)dt2 t e µt x u 2ν (x,s)q ν (x,t s)dsdxdt. (3.26) Since p ν (t) appears as the result of the integral of probability densities, we can conclude that p ν (t) for all and t>. We provide an alternative proofof the non-negativityof p ν (t), t>, and of pν (t), based on the representation of the fractional linear birth process N ν (t) as N ν (t)n(t 2ν (t)), <ν, (3.27)

16 Fractional birth processes 873 where T 2ν (t) possesses a distribution coinciding with the folded solution of the fractional diffusion equation 2ν u t 2ν 2 u x2, <ν, (3.28) u(x,)δ(x), with the further condition that u t (x,) for /2<ν. Theorem 3.3. The probability generating function G ν (u,t)eu Nν(t) of N ν (t), t>, has the Laplace transform e µt ue t G ν (u,t)dt u( e t ) µν e µνt dt. (3.29) Proof. We evaluate the Laplace transform (3.29) as follows: e µt G ν (u,t)dt e µt u ( ) j E j ν, ( jt ν )dt u µν µν uµν uµν uµν j l j ( j u u l uµν j j µν )( ) µ ν +j ( ) j j µ ν /++j ()! (µ ν /+)(µ ν /+2) (µ ν /+) u l l! (µ ν /+) (µ ν /++l) ) u l B (l+, µν + u l x l ( x) µν / dx l ( x) µν / ( ux) dx( xe t ) ue t ν u( e t ) e tµ µ ν dt. (for <ux<) (by (3.7)) (3.3)

17 874 E. Orsingher and F. Polito Remar 3.3. In order to extract from (3.29) the representation (3.27), we note that { } e µt u Pr{N(T 2ν (t))} dt e µt { } u Pr{N(s)}f T2ν (s,t)ds dt (3.3) G(u,s)µ ν e µνs ds, which coincides with (3.29). It can be shown that is the Laplace transform of the folded solution to e µt f T2ν (s,t)dtµ ν e sµν, s>, (3.32) 2ν u t 2ν 2 u s2, <ν, (3.33) with the initial condition u(s,)δ(s) for <ν and also u t (s,) for /2<ν. In the light of (3.27), the non-negativity of p ν (t) is immediate because Pr{N ν (t)} Pr{N(s)}Pr{T 2ν (t) ds}. (3.34) The relation (3.34) immediately leads to the conclusion that Pr{N ν(t)}. Some explicit expressions for (3.34) can be given when the Pr{T 2ν (t) ds} can be wored out in detail. We now that for ν /2 n, we have that Pr{T /2 n (t) ds} Pr{ B ( B 2 ( B n (t) ) ) ds} (3.35) ds2 n e s2 /(4ω ) 4πω dω e ω2 /(4ω2) 4πω2 dω 2 e ω2 n /(4t) 4πt dω n. For details concerning (3.35), see Theorem 2.2 of Orsingher and Beghin [], where the differences of the constants depend on the fact that the diffusion coefficient in equation (3.33) equals instead of 2 (/2n ) 2. The distribution (3.35) represents the density of the folded (n )-times iterated Brownian motion and therefore B,...,B n are independent Brownian motions with volatility equal to 2.

18 Fractional birth processes 875 For ν /3, the process (3.27) has the form N /3 (t) N( A(t) ), where A(t) is a process whose law is the solution of 2/3 u t 2 u 2/3 x2, u(x,)δ(x). (3.36) In Orsingher and Beghin [], it is shown that the solution to (3.36) is where u 2/3 (x,t) 3 2 A i (x) π 3 3t A i ( x 3 3t ), (3.37) ) cos (αx+ α3 dα (3.38) 3 is the Airy function. Therefore, in this case, the distribution (3.34) has the form p /3 (t) e s ( e s ) 3 3 3t A i ( s 3 3t )ds,,t>. (3.39) Remar 3.4. From (3.3), it is straightforward to show that the probability generating function G ν (u,t)eu Nν(t) satisfies the partial differential equation ν t νg(u,t)u(u ) G(u,t), <ν, u (3.4) G(u,)u, and thus EN ν (t) G u u is the solution to d ν dt νen ν EN ν, <ν, EN ν (). (3.4) The solution of (3.4) is EN ν (t)e ν, (t ν ), t>. (3.42) Clearly, the result (3.42) can be also derived by evaluating the Laplace transform { } e µt EN ν (t)dt e µt Pr{N(s)}Pr{T 2ν (t) ds} dt e µt e s Pr{T 2ν (t) ds}dt e s µ ν e sµν ds µν µ ν e µt E ν, (t ν )dt

19 876 E. Orsingher and F. Polito and this verifies (3.42). The mean value (3.42) can be obtained in a third manner: e µt EN ν (t) µν µν µν µν j j ( )( ) j E j ν, ( jt ν )e t dt ( j µν )( ) j µ ν +j ( ) j j µ ν /++j j ()! (µ ν /+) (µ ν /+) Γ()Γ(µν /+) Γ(µ ν /++) x ( x) µν / µν µ ν (3.43) e µt E ν, (t ν )dt. The result of Remar 3.4, EN ν (t) E ν, (t ν ), should be compared with the results of Uchaiin, Cahoy and Sibatov [6]. An interesting representation of (3.42) following from (3.27) gives that EN ν (t) e s Pr{T 2ν (t) ds} EN(s)Pr{T 2ν (t) ds}. (3.44) The expansion of the population subject to the law of the fractional birth process is increasingly rapid as the order of fractionality ν decreases. This is shown in Figure and this behavior is due to the increasing structure of the gamma function for ν > appearing in the Mittag Leffler function E ν,. This qualitative feature of the process being investigated here shows that it conveniently applies to explosively expanding populations. Remar 3.5. By twice deriving (3.4) with respect to u, we obtain the fractional equation for the second-order factorial moment E{N ν (t)(n ν (t) )}g ν (t), (3.45) that is, ν t νg ν(t)2g ν (t)+2en ν (t), <ν, g ν (). (3.46)

20 Fractional birth processes 877 The Laplace transform of the solution to (3.46) is 2µ ν H ν (t) e µt g ν (t)dt (µ ν )(µ ν 2) { 2µ ν µ ν 2 } µ ν. (3.47) The inverse Laplace transform of (3.47) is E{N ν (t)(n ν (t) )}2E ν, (2t ν ) 2E ν, (t ν ). (3.48) It is now straightforward to obtain the variance from (3.48), VarN ν (t)2e ν, (2t ν ) E ν, (t ν ) E 2 ν, (tν ). (3.49) For ν, we retrieve from (3.49) the well-nown expression of the variance of the linear birth process VarN (t)e t (e t ). (3.5) Figure. Mean number of individuals at time t for various values of ν.

21 878 E. Orsingher and F. Polito Remar 3.6. IfX,...,X n arei.i.d.randomvariableswithcommondistributionf(x) Pr(X < x), then we can write the following probability: Pr{max(X,...,X Nν(t))<x} (Pr{X <x}) Pr{N ν (t)} (by (3.27)) Analogously, we have that G(F(x),s)Pr{T 2ν (t) ds} F(x)e s F(x)( e s ) Pr{T 2ν(t) ds}. Pr{min(X,...,X Nν(t))>x} ( F(x))e s ( F(x))( e s ) Pr{T 2ν(t) ds}. (3.5) (3.52) Remar 3.7. If the initial number of components of the population is n, then the p.g.f. becomes E(u Nν(t) N ν ()n ) u +n e zn ( n + ) ( e z ) Pr{T 2ν (t) dz}. From (3.53), we can extract the distribution of the population size at time t as (3.53) Pr{N ν (t)+n N ν ()n } n + e zn ( e z ) Pr{T 2ν (t) dz}, If we write +n, then we can rewrite (3.54) as. (3.54) Pr{N ν (t) N ν ()n } e zn ( e z ) n Pr{T n 2ν (t) dz}, n, (3.55) where is the number of individuals in the population at time t. For n, formulas (3.54), (3.55) coincide with (3.4). The random time T 2ν (t), t>, appearing in (3.54) and (3.55) has a distribution which is related to the fractional equation 2ν u t 2ν 2 u z2, <ν. (3.56)

22 Fractional birth processes 879 It is possible to slightly change the structure of formulas (3.54) and (3.55) by means of the transformation z y so that the distribution of T 2ν (t) becomes related to the equation 2ν u u t 2ν 2 2 y2, <ν, (3.57) where (3.) shows the connection between the diffusion coefficient in (3.57) and the birth rate. Remar 3.8. If we assume that the initial number of individuals in the population is N ν () n, then we can generalize the result (3.4) offering a representation of the distribution of N ν (t) alternative to (3.55). If we tae the Laplace transform of (3.55), then we have that e µt Pr{N ν (t)+n N ν ()n }dt n + e zn ( e z ) Pr{T 2ν (t) dz}dt n + e zn ( e z ) µ ν e µνz dz ( n + )µ ν e z(n+µν) ( e z ) dz ( n + ( n + ) µ ν r ) µ ν r ( )( ) r e z(n+r+µν) dz r ( ) r r (n +r)+µ ν. (by(3.32)) (3.58) By taing the inverse Laplace transform of (3.58), we have that Pr{N ν (t)+n N ν ()n } n + ( ) r E r ν, ( (n +r)t ν ). r (3.59) From (3.59), we can infer the interesting information Pr{N ν (dt)n + N ν ()n } ( n ( ) r) r E ν, ( (n +r)(dt) ν ) (3.6) r

23 88 E. Orsingher and F. Polito n [E ν, ( n (dt) ν ) E ν, ( (n +)(dt) ν )] n (dt) ν Γ(ν +) by writing only the lower order terms. This shows that the probability of a new offspring at the beginning of the process is proportional to (dt) ν and to the initial number of progenitors. From our point of view, this is the most important qualitative feature of our results since it maes explicit the dependence on the order ν of the fractional birth process. Theorem 3.4. The Laplace transform of the probability generating function G ν (t,u) of the fractional linear birth process has the form H ν (µ,u) e µt G ν (t,u)dt uµν ( x) µν / xu dx, <u<, µ>. (3.6) Proof. We saw above that the function G ν solves the Cauchy problem ν G ν t ν u(u ) G ν, <ν, u G ν (u,)u. (3.62) By taing the Laplace transform of (3.62), we have that µ ν H ν µ ν uu(u ) H ν u. (3.63) By inserting (3.6) into (3.63) and performing some integrations by parts, we have that uµ 2ν ( x) µν / dx uµ ν xu [ µ ν ( x) µν / u(u ) xu [ µ ν ( x) µν / u(u ) xu u(u )µ2ν µν uµ ν + uµ2ν ( x) µν / ( xu) x( x) µν / dx ( xu) ( x) µν / ( xu) and this concludes the proof of Theorem 3.4. dx+ uµν dx+ µν dx+ µ2ν 2 dx, ( x) µν / ] x ( xu) 2 dx x( x) µν / xu x x x( x) µν / ( xu) ] dx (3.64)

24 Fractional birth processes 88 Remar 3.9. We note that H ν (µ,u) u /µ because G ν (t,). Furthermore, H ν (µ,u) u µν u µ ν e µt E ν, (t ν )dt, (3.65) which accords well with (3.42). Acnowledgement The authors are pleased to acnowledge the remars of an unnown referee which improved the quality of this paper. References [] Bartlett, M.S. (978). An Introduction to Stochastic Processes, with Special Reference to Methods and Applications, 3rd ed. Cambridge: Cambridge Univ. Press. MR [2] Beghin, L. and Orsingher, E. (29). Fractional Poisson processes and related planar random motions. Electron. J. Probab MR25354 [3] Cahoy, D.O. (27). Fractional Poisson processes in terms of alpha-stable densities. Ph.D. thesis. [4] Feller, W. (968). An Introduction to Probability Theory and Its Applications, Volume, 3rd ed. New Yor: Wiley. MR2282 [5] Gihman, I.I. and Sorohod, A.V.(996). Introduction to the Theory of Random Processes. New Yor: Dover Publications. MR4355 [6] Gradshteyn, I.S. and Ryzhi, I.M. (98). Table of Integrals, Series, and Products. New Yor: Academic Press. MR [7] Jumarie, G. (2). Fractional master equation: Non-standard analysis and Liouville Riemann derivative. Chaos Solitons Fractals MR8579 [8] Kirschenhofer, P. (996). A note on alternating sums. Electron. J. Combin. 3. MR [9] Lasin, N. (23). Fractional Poisson process. Commun. Nonlinear Sci. Numer. Simul MR273 [] Orsingher, E. and Beghin, L. (24). Time-fractional telegraph equations and telegraph processes with Brownian time. Probab. Theory Related Fields MR [] Orsingher, E. and Beghin, L. (29). Fractional diffusion equations and processes with randomly-varying time. Ann. Probab MR [2] Podlubny, I. (999). Fractional Differential Equations. San Diego: Academic Press. MR65822 [3] Repin, O.N. and Saichev, A.I. (2). Fractional Poisson law. Radiophys. and Quantum Electronics MR934 [4] Samorodnitsy, G. and Taqqu, M.S. (994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. New Yor: Chapman and Hall. MR28932 [5] Uchaiin, V.V. and Sibatov, R.T. (28). A fractional Poisson process in a model of dispersive charge transport in semiconductors. Russian J. Numer. Anal. Math. Modelling MR244873

25 882 E. Orsingher and F. Polito [6] Uchaiin, V.V., Cahoy, D.O. and Sibatov, R.T. (28). Fractional processes: From Poisson to branching one. Int. J. Bifurcation Chaos MR [7] Wang, X.-T. and Wen, Z.-X. (23). Poisson fractional processes. Chaos Solitons Fractals MR [8] Wang, X.-T., Wen, Z.-X. and Zhang, S.-Y. (26). Fractional Poisson process (II). Chaos Solitons Fractals MR [9] Wang, X.-T., Zhang, S.-Y. and Fan, S. (27). Nonhomogeneous fractional Poisson processes. Chaos Solitons Fractals MR Received November 28 and revised August 29