Orsingher, E.

Polito, Federico

English

Institutional Research Information System University of Turin

XIII International EM Conference on Eventological Mathematics and Related Fields, Krasnoyarsk, Siberia, Russia, 2009Some results on time-varying fractional partial differential equationsand birth-death processesEnzo OrsingherDipartimento di Statistica, Probabilita` e Stat. Appl.“Sapienza” Universita` di RomaRome, Italyenzo.orsingher@uniroma1.itFederico PolitoDipartimento di Statistica, Probabilita` e Stat. Appl.“Sapienza” Universita` di RomaRome, Italyfederico.polito@uniroma1.itAbstract. Some results on specific types of branchingprocesses are presented. Firstly, linear pure birth andbirth-death processes governed by partial differentialequations with time-varying coefficients are analysed.Such processes are constructed by inserting thefractional time derivative into the p.d.e. governing thelaw of fractional Brownian motion. We consider alsopure birth processes stopped at first-passage time ofBrownian motion and present the related distributionsand the governing equations. Some explicit results onthe mean values and low-order probabilities are obtainedin terms of generalised Mittag-Leffler functions.Keywords. Fractional derivatives, branching processes,partial differential equations with time-varyingcoefficients, processes with random time.1 IntroductionIn this paper we consider compositions of differenttypes of processes whose distribution is related eitherto fractional equations or to higher-order partialdifferential equations. In a previous paper of ours [4]we considered the fractional pure birth process Nν(t),t > 0 whose state probabilitiespˆνk(t) = Pr {Nν(t) = k} , (1)where k ≥ 1 (with one initial progenitor), t > 0, and ν ∈(0, 1], satisfy the time-fractional difference-differentialequationsdνdtνpk(t) = −λkpk(t) + λk−1pk−1(t), (2)subject to the initial conditionspk(0) ={1, k > 10, k = 1. (3)For λk = λk, k ≥ 1 we have the linear pure birthprocess. We have obtained the explicit distributionin the linear and non-linear cases with many relatedproperties of the fractional birth process. We have alsoshown that Nν(t), t > 0 can be represented asNν(t) = N (T2ν(t)) (4)where N(t), t > 0, is the classical birth process andT2ν(t) is a stochastic process independent from N(t)and possessing distribution related to the solution ofthe Cauchy problem∂2ν∂t2νf(x, t) =∂2∂x2f(x, t), ν ∈ (0, 1]f(x, 0) = δ(x). (5)The fractional derivative appearing in (2) and (5) mustbe understood as followsdνh (t)dtν=∫ t0ddsh (s)(t− s)ν dsΓ (1− ν) , ν ∈ (0, 1)h′ (t) , ν = 1. (6)We consider here the equationdνdtνpk(t) = λt2H−1 {(k − 1)pk−1(t)− kpk(t)} (7)which generalises (2) and produces for the probabilitygenerating function G(u, t) = EuNν(t) the p.d.e.∂ν∂tνG(u, t) = λt2H−1u(u− 1) ∂∂uG(u, t) (8)subject to the initial condition G(u, 0) = u. Thetime-dependent coefficient in (8) is inspired by thestructure of the partial differential equation governingthe distribution g(t, x) of the fractional Brownianmotion which reads∂∂tg(t, x) = Ht2H−1∂2∂x2g(t, x) (9)with the initial condition g(0, x) = δ(x), and coincideswith the classical heat equation for H = 1/2. From(4) the process N 12(t), t > 0, has the followingrepresentationN 12(t) = N (|B(t)|) (10)24 XIII International EM’2009 Conferencewhere B(t), t > 0 is a standard Brownian motionindependent from N(t). The process related to (7) and(8) has the representationN 12(t) = N (|BH(t)|) (11)where BH(t), t > 0 is a fractional Brownian motionindependent from N(t), t > 0. We also study in detailother properties and this involves a delicate analysisbased on the generalised Mittag-Leffler functions of theformEα,m,l(z) = 1+∞∑k=1zkk−1∏j=0Γ (α (jm + l) + 1)Γ (α (jm + l + 1) + 1). (12)2 Partial differential equations withtime-varying coefficients andbranching processesIt is well-known that the partial differential equationsatisfied by the transition law of the fractional Brownianmotion is a simple time-modification of the heatequation (see appendix 6). We are interested inanalysing similar differential equations. An example ofsuch equations is the analogue of that governing aclassical linear birth process (see for example [2], page87) but with time-varying coefficients i.e.∂∂tG(u, t) = λt2H−1u(u− 1) ∂∂uG(u, t)G(u, 0) = u, (13)where t > 0, H ∈ (0, 1] and λ is the birth rate. ByG = G(u, t), 0 < u ≤ 1, t > 0, we denote the probabilitygenerating function of the pure birth process related to(7) for ν = 1. From (13) we infer that for the mean valueEN(t) = Gu(u, t)|u=1 we arrive at the Cauchy problemddtEN(t) = λt2H−1EN(t)EN(0) = 1, (14)where t > 0, H ∈ (0, 1] and λ is the rate of birth.Equation (14) is readily solved by standard methodsand yields EN(t) = eλ2Ht2H .With similar arguments it is possible to solve also thefollowing difference-differential equationddtp1(t) = −λt2H−1p1(t)p1(0) = 1, (15)with t > 0, H ∈ (0, 1] and where p1(t) is the probabilityof having exactly one individual at time t given that attime t = 0 we start with one individual, i.e. p1(t) =Pr {N(t) = 1 |N(0) = 1}. The solution to (15) readsp1(t) = e− λ2Ht2H .It can be checked by direct calculations that probabilitygenerating function has the formG(u, t) =ue−λ2Ht2H1− u(1− e− λ2H t2H) , t > 0. (16)This implies that the distribution of N(t), t > 0 ispk(t) = e−λt2H(1− e−λt2H)k−1, k ≥ 1. (17)Remark 2.1 As in the case of (44) (see appendix 6)when H = 1/2 we obtain, as a particular case, theclassical linear pure birth process.A second example worth of being analysed is thatof a linear birth-death process governed by partialdifferential equations with time-varying coefficients. Fordetails about the classical case see [2], page 93. Here wemodify the p.d.e. involving the probability generatingfunction exactly in the same manner as in (13), thusobtaining∂∂tG(u, t) = t2H−1(λu− µ)(u− 1) ∂∂uG(u, t)G(u, 0) = u,(18)where t > 0, H ∈ (0, 1] and where λ and µ arerespectively the birth rate and the death rate. Again,by means of standard methods it can be shown that thefunctionG(u, t) = (u− 1) µeλ−µ2Ht2H − λu + µ(u− 1)λeλ−µ2H t2H − λu + µ(19)is the solution to (18).Remark 2.2 The probability generating functionreduces to that of the classical case when H = 1/2.From the probability generating function it is simple toextract the extinction probability p0(t) which readsp0(t) = G(0, t) = µ− µeλ−µ2Ht2Hµ− λeλ−µ2H t2H. (20)We also note that the behaviour of p0(t), when t tendsto infinity, is the same as in the classical case and doesnot depend on the value of H.3 Bifractional pure growth processIn order to furnish the system with a further type offractionality, we substitute the integer time-derivativeOrsingher and Polito 25in (13) with a fractional derivative defined in (6).This kind of fractional derivative (usually calledCaputo or Dzhrbashyan-Caputo fractional derivative) isclosely related to the usual Riemann-Liouville fractionalderivative (for details see [1], page 90). We thereforeobtain the following fractional p.d.e.∂ν∂tνG(u, t) = λt2H−1u(u− 1) ∂∂uG(u, t)G(u, 0) = u, (21)where ν ∈ (0, 1], t > 0, H ∈ (0, 1], G(u, t) = EuN(t),0 < u ≤ 1 and the fractional derivative is defined in(6). From (21) we can extract some information onthe expectation of the process N(t), t > 0 which hasprobability generating function G(u, t). By consideringthat∂G∂u∣∣∣∣u=1= EN(t), t > 0, ν ∈ (0, 1], (22)we obtain dνdtνEN(t) = λt2H−1EN(t)EN(0) = 1, (23)where t > 0, ν ∈ (0, 1] and H ∈ (0, 1]. The solution to(23) is directly obtained from [1], page 233 asN(t) = Eν,1+ 2H−1ν, 2H−1ν(λtν+2H−1), t > 0, (24)where Eα,m,l(z) is the generalised Mittag-Lefflerfunction defined asEα,m,l(z) = 1+∞∑k=1zkk−1∏j=0Γ (α (jm + l) + 1)Γ (α (jm + l + 1) + 1), (25)and with (1−ν)/2 < H ≤ 1. The solution is then provedto be unique when 1/2 ≤ H ≤ 1. For details see theorem6.2.We now address the problem of determining theprobability law of the process N(t), t > 0 i.e. pνk(t) =Pr {N(t) = k |N(0) = 1} for k = 1. We must solve thefractional difference-differential equationdνdtνp1(t) = −λt2H−1p1(t),p1(0) = 1, (26)where t > 0, ν ∈ (0, 1] and H ∈ (0, 1]. Similarly to thesolution to (23), the solution to (26) ispνk(t) = Eν,1+ 2H−1ν, 2H−1ν(−λtν+2H−1) , (27)with again, t > 0, (1 − ν)/2 < H ≤ 1 and where thesolution is proved to be unique for 1/2 ≤ H ≤ 1.Remark 3.1 From respectively (27) and (24), whenν = 1, we retrieve the results in section 2. WhenH = 1/2, they reduce to the corresponding expressionsfor the fractional linear pure birth process (see [4]).4 Bifractional diffusionWe consider here a generalised version of the so-calledtime-fractional diffusion equation (see e.g. [3]) as follows∂ν∂tνu (x, t) = Ht2H−1∂2∂x2u (x, t)u (x, 0) = δ (x), (28)with ν ∈ (0, 2], x ∈ R, t ∈ R+, together with the furthercondition ut(x, t)|t=0 = 0 when ν ∈ (1, 2] The fractionalderivative appearing in (28) is the Dzhrbashyan-Caputofractional derivative (6) but with ν ∈ (0, 2] and theparameter H ∈ (0, 1].Theorem 4.1 Let u(x, t), x ∈ R, t ∈ R+ be thesolution to the generalised time-fractional diffusionequation (28). The function q(x, t) can be written asq(x, t) = (29)=12pi∫ +∞−∞e−iβxEν,1+ 2H−1ν, 2H−1ν(−Hβ2tν+2H−1) ,where t > 0, x ∈ R, H ∈ (0, 1] and ν ∈ (0, 2].Proof. By resorting to the Fourier transform of (28) weobtain the following differential equation∂ν∂tνU (β, t) = −Ht2H−1β2U (β, t)U(β, 0) = 1, (30)where t > 0, H ∈ (0, 1], ν ∈ (0, 2] and with theadditional condition Ut(β, t)|t=0 = 0 when ν ∈ (1, 2].Equation (30) is solved by (see [1], page 233, examples4.11 and 4.12)U(β, t) = Eν,1+ 2H−1ν, 2H−1ν(−Hβ2tν+2H−1) , (31)with (1 − ν)/2 < H ≤ 1. The solution is proved tobe unique for 1/2 ≤ H ≤ 1. Note that the functionEα,m,l(z) is the generalised Mittag-Leffler (25). From(31) it is straightforward to obtain (29).5 Birth processes withrandomly-varying timeLet q(t, s) be the transition density of the first-passagetime Tt at time t of a Brownian motion i.e.q(t, s) =te−t22s√2pis3ds, (t, s) ∈ R+× R+. (32)26 XIII International EM’2009 ConferenceIt is well-known that q = q(t, s) is a solution to thepartial differential equation∂2∂t2q(t, s) = 2∂∂sq(t, s), (t, s) ∈ R +×R+. (33)We are interested in composing a classical pure birthprocess N(t), t > 0, with the first-passage time Tt, t > 0.We have thatp˘k(t) = Pr {N (Tt) = k|N(0) = 1} (34)=∫ ∞0pk(t)q(t, s)ds.Theorem 5.1 Let N(t), t > 0 be a classical pure birthprocess and Tt, t > 0 be the first-passage time at tof a standard Brownian motion. The state probabilitiesp˘k(t) = Pr {N(Tt) = k|N(0) = 1} satisfy the followingdifference-differential equationd2dt2p˘k(t) = 2λkp˘k(t)− 2λk−1p˘k−1(t), (35)where t > 0 and k ≥ 1.Proof. Result (35) is directly obtained by evaluating thesecond order derivative as followsd2dt2p˘k(t) =∫ ∞0pk(s)∂2∂t2q(t, s)ds (36)= 2∫ ∞0pk(s)∂∂sq(t, s)ds= 2q(t, s)pk(s)|∞s=0 − 2∫ ∞0ddspk(s)q(t, s)ds= −2∫ ∞0q(t, s) {−λkpk(s) + λk−1pk−1(s)} ds= 2λkp˘k(t)− 2λk−1p˘k−1(t)where pk(t), t > 0, k ≥ 1, are the state probabilitiesof a classical pure birth process. Note that we appliedequation (33) in the second step of the proof.The state probabilities p˘k(t), t > 0, can also be writtenexplicitly. This is shown for the linear case in the nexttheorem (λ is the birth rate).Theorem 5.2 The distribution of the linear pure birthprocess stopped at a random first-passage time Tt ofa standard Brownian motion with a simple initialprogenitor readsp˘k(t) =k∑m=1(k − 1m− 1)(−1)m−1e−t√2λm, (37)with k ≥ 1 and t > 0.Proof.p˘k(t) = Pr {N(Tt)|N(0) = 1} (38)=∫ ∞0e−λs(1− e−λs)k−1 te−t22s√2pis3ds=k−1∑m=0(k − 1m)∫ ∞0(−e−λs)m e−λs te−t22s√2pis3ds=k−1∑m=0(k − 1m)(−1)m∫ ∞0e−λs(1+m)te−t22s√2pis3ds=k−1∑m=0(k − 1m)(−1)me−t√2λ(1+m)=k∑m=1(k − 1m− 1)(−1)m−1e−t√2λm.This concludes the proof.In the next theorem we prove that the distribution (37)is actually a solution of the p.d.e. (35).Theorem 5.3 Let N(t), t > 0 be a linear pure birthprocess and Tt, t > 0 the first-passage time process ofstandard Brownian motion. The state probabilities ofthe composed process N(Tt), t > 0, i.e.p˘k(t) =k∑m=1(k − 1m− 1)(−1)m−1e−t√2λm, (39)where k ≥ 1, t > 0, satisfy the following difference-differential equationsd2dt2p˘k(t) = 2λkp˘k(t)− 2λk−1p˘k−1(t), (40)where k ≥ 1, t > 0.Proof. It is sufficient to calculate the left hand side andright hand side of (40) and show that they coincide. Forthe left hand side we haved2dt2k∑m=1(k − 1m− 1)(−1)m−1e−t√2λm = (41)=k∑m=1(k − 1m− 1)(−1)m−1(2λm)e−t√2λm.Orsingher and Polito 27For the right hand side we havekp˘k(t)− (k − 1)p˘k−1(t) = (42)= kk∑m=1(k − 1m− 1)(−1)m−1e−t√2λm+− (k − 1)k−1∑m=1(k − 2m− 1)(−1)m−1e−t√2λm=k−1∑m=1(−1)m−1e−t√2λm[(k − 1)!(m− 1)! (k −m− 1)!××(kk −m − 1)]+ k(−1)k−1e−t√2λk=k−1∑m=1(−1)m−1e−t√2λm(m (k − 1)!(m− 1)! (k −m)!)++ k(−1)k−1e−t√2λk=k∑m=1(k − 1m− 1)(−1)m−1e−t√2λmm,and this concludes the proof.6 AppendixIt is possible to obtain the p.d.e. satisfied by theprobability density function of the fractional Brownianmotion BH(t), z ∈ R+ by a simple time-scalingprocedure as followsTheorem 6.1 Let u(z, x) be the transition density of astandard Brownian motion satisfying the heat equation∂∂zu(z, x) =12∂2∂x2u(z, x) (43)together with the usual initial condition u(0, x) = δ(x).After the substitution z = t2H , H ∈ (0, 1], u(z, x) =g(t, x) satisfies the following partial differential equationwith time-varying coefficients∂∂tg(t, x) = Ht2H−1∂2∂x2g(t, x) (44)with the initial condition g(0, x) = δ(x).Proof. By evaluating the partial derivative of g(t, x)with respect to time t we have∂∂tg(t, x) =∂∂tu(z, x) =∂∂zu(z, x)2Ht2H−1 (45)= Ht2H−1∂2∂x2u(z, x)= Ht2H−1∂2∂x2g(t, x)thus obtaining (44).In the following theorem we prove that the generalisedMittag-Leffler function (25) is the solution to thedifferential equation considered in section 3.Theorem 6.2 The generalised Mittag-Leffler functionEα,1+ βα,βα(λtα+β)= 1 +∞∑k=1λktk(α+β)× (46)×k−1∏j=0Γ(α(j + j βα+ βα)+ 1)Γ(α(j + j βα+ βα+ 1)+ 1)solves the ordinary differential equation with variablecoefficients dαdtαy (t) = λtβy (t)y (0) = 1, (47)where t ∈ R+, α ∈ (0, 1], β > 0 and where dα/dtα isthe Dzhrbashyan-Caputo fractional derivative (6).Proof. The theorem is simply proved by evaluating thefractional derivative in the left hand side of (47) asfollowsdαdtαy(t) = (48)=∞∑k=1λkk−1∏j=0Γ (αj + βj + β + 1)Γ (αj + βj + β + α + 1)dαdtα[tk(α+β)]=∞∑k=1λkΓ (kα + kβ + 1)Γ (kα + kβ + 1− α) tk(α+β)−α××k−1∏j=0Γ (αj + jβ + β + 1)Γ (αj + βj + β + α + 1)= λtβ∞∑r=0λrtr(α+β)Γ (r (α + β) + α + β + 1)Γ (r (α + β) + β + 1)××r∏j=0rΓ (αj + jβ + β + 1)Γ (αj + βj + β + α + 1)= λtβ1 +∞∑r=1λrtr(α+β)r−1∏j=0Γ (αj + βj + β + 1)Γ (αj + βj + β + α + 1) .References[1] M. Srivastava H A. A. Kilbas and J. J. Trujillo. Theory andapplications of fractional differential equations. Elsevier Science,2006-02.[2] Norman Bailey. The Elements of Stochastic Processes withApplications to the Natural Sciences. John Wiley & Sons, NewYork, 1964.[3] Yasuhiro Fujita. Integrodifferential equation which interpolatesthe heat equation and the wave equation. J. Math.,27(2):309Џ321, 1990.[4] Enzo Orsingher and Federico Polito. Fractional pure birthprocesses. To appear in Bernoulli, 2009.

Some results on time-varying fractional partial differential equations and birth-death processes

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Some results on time-varying fractional partial differential equations and birth-death processes

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