Mathematics Subject Classification: 26A33, 76M35, 82B31A stochastic solution is constructed for a fractional generalization of
the KPP (Kolmogorov, Petrovskii, Piskunov) equation. The solution uses
a fractional generalization of the branching exponential process and propagation
 processes which are spectral integrals of Levy processes

Cipriano, F.

Ouerdiane, H.

Vilela Mendes, R.

Fractional Calculus and Applied Analysis

Bulgarian Digital Mathematics Library at IMI-BAS

STOCHASTIC SOLUTION OF A KPP-TYPE NONLINEARFRACTIONAL DIFFERENTIAL EQUATIONF. Cipriano 1, H. Ouerdiane 2, R. Vilela Mendes 3AbstractA stochastic solution is constructed for a fractional generalization ofthe KPP (Kolmogorov, Petrovskii, Piskunov) equation. The solution usesa fractional generalization of the branching exponential process and propa-gation processes which are spectral integrals of Levy processes.Mathematics Subject Classification: 26A33, 76M35, 82B31Key Words and Phrases: fractional partial differential equations, stochas-tic solutions1. Introduction: The notion of stochastic solutionThe solutions of linear elliptic and parabolic equations, both with Cauchyand Dirichlet boundary conditions, have a probabilistic interpretation. Theseare classical results which may be traced back to the work of Courant,Friedrichs and Lewy [1] in the 1920’s and became a standard tool in poten-tial theory [2],[3]. For example, for the heat equation,∂tu(t, x) =12∂2∂x2u(t, x) with u(0, x) = f(x), (1)the solution may be written either asu (t, x) =12√pi∫1√texp(−(x− y)22t)f (y) dy, (2)or as48 F. Cipriano, H. Ouerdiane, R. Vilela Mendesu(t, x) = Exf(Xt), (3)where Ex means the expectation value, starting from x, of the processdXt = dWt,Wt being the Wiener process.Eq.(1) is a specification of a problem whereas (2) and (3) are solutions inthe sense that they both provide algorithmic means to construct a functionsatisfying the specification. An important condition for (2) and (3) to beconsidered as solutions is the fact that the algorithmic tools are independentof the particular solution, in the first case an integration procedure and inthe second the simulation of a solution-independent process. This shouldbe contrasted with stochastic processes constructed from a given particularsolution, as has been done for example for the Boltzman equation [4].In contrast with the linear problems, for nonlinear partial differentialequations, explicit solutions in terms of elementary functions or integralsare only known in very particular cases. However, if a solution-independentstochastic process is found that (for arbitrary initial conditions) generatesthe solution in the sense of Eq.(3), a stochastic solution is obtained. Inthis way the set of equations for which exact solutions are known would beconsiderably extended.The stochastic representations recently constructed for the Navier-Stokesequations ([5], [6], [7], [8]) and for the Vlasov-Poisson equations ([9], [10]) de-fine solution-independent processes for which the mean values of some func-tionals are solutions to these equations. Therefore, they are exact stochasticsolutions.In the stochastic solutions one deals with a process that starts from thepoint where the solution is to be found, a functional being then computedalong the whole sample path or until it reaches a boundary. In all casesone needs to average over many independent sample paths to obtain a ex-pectation value of the functional. The localized and parallelizable nature ofthe solution construction is clear. Provided some differentiability conditionsare satisfied, the process also handles equally well simple or very complexboundary conditions.The stochastic solutions also provide an intuitive characterization ofthe physical phenomena, relating nonlinear interactions with cascading pro-cesses. By the study of exit times from a domain they also sometimes pro-vide access to quantities that cannot be obtained by perturbative methods,see [11].STOCHASTIC SOLUTION OF A KPP-TYPE NONLINEAR . . . 49One way to construct stochastic solutions is based on a probabilistic in-terpretation of the Picard series. The differential equation is written as anintegral equation which is rearranged in a such a way that the coefficientsof the successive terms in the Picard iteration obey a normalization condi-tion. The Picard iteration is then interpreted as an evolution and branchingprocess, the stochastic solution being equivalent to importance sampling ofthe normalized Picard series. This method is used in this paper to obtaina stochastic solution of a nonlinear partial differential equation, which is afractional version of the Kolmogorov-Petrovskii-Piskunov (KPP) equation,see [12].2. A fractional nonlinear partial differential equationWe consider the following equationtDα∗ u (t, x) =12 xDβθ u (t, x) + u2 (t, x)− u (t, x) . (4)We use the same notations as in the study of the linear problem in [13].here tDα∗ is a Caputo derivative of order α,tDα∗ f (t) =1Γ (m− α)∫ t0f (m) (τ) dτ(t− τ)α+1−m m− 1 < α < m,dmdtmf (t) α = m,(5)with m integer, and xDβθ is a Riesz-Feller derivative defined through itsFourier symbol byF{xDβθ f (x)}(k) = −ψθβ (k)F {f (x)} (k) , (6)with ψθβ (k) = |k|β ei(signk)θpi/2.Eq. (4) is a fractional version of the KPP equation, studied by proba-bilistic means by McKean [14]. Physically it describes a nonlinear diffusionwith growing mass and in our fractional generalization it would representthe same phenomenon taking into account memory effects in time and longrange correlations in space.As outlined in the introduction, the first step towards a probabilisticformulation is the rewriting of Eq. (4) as an integral equation including theinitial conditions. For this purpose we take the Fourier transform (F) inspace and the Laplace transform (L) in time obtaining50 F. Cipriano, H. Ouerdiane, R. Vilela Mendessα˜̂u (s, k) (s, k) = sα−1û (0+, k)− 12ψθβ (k) ˜̂u (s, k)− ˜̂u (s, k)+∫ ∞0dte−stF (u2 (t, x)) , (7)whereû (t, k) = F (u (t, x)) =∫ ∞−∞eikxu (t, x) dx,u˜ (s, x) = L (u (t, x)) =∫ ∞0e−stu (t, x) dt.This equation holds for 0 < α ≤ 1 or for 0 < α ≤ 2 with ∂dtu (0+, x) = 0.Solving for ˜̂u (s, k) one obtains an integral equation˜̂u (s, k) = sα−1sα + 1 + 12ψθβ (k)û(0+, k)+∫ ∞0dte−stsα + 1 + 12ψθβ (k)F (u2 (t, x)) .(8)Taking the inverse Fourier and Laplace transforms (see e.g. [15]), we haveu (t, x) = Eα,1 (−tα)∫ ∞−∞dyF−1Eα,1(−(1 + 12ψθβ (k))tα)Eα,1 (−tα)× (x− y)u (0+, y)+ ∫ t0dτ (t− τ)α−1Eα,α (− (t− τ)α)∫ ∞−∞dyF−1Eα,α(−(1 + 12ψθβ (k))(t− τ)α)Eα,α (− (t− τ)α) (x− y)u2 (τ, y) , (9)where Eβ,ρ is the generalized Mittag-Leﬄer function (see e.g. [15])Eα,ρ (z) =∞∑j=0zjΓ (αj + ρ).We define the following propagation kernelGβα,ρ (t, x) = F−1Eα.ρ(−(1 + 12ψθβ (k))tα)Eα,ρ (−tα) (x) (10)and from the normalization relation,STOCHASTIC SOLUTION OF A KPP-TYPE NONLINEAR . . . 51Eα,1 (−tα) +∫ t0dτ (t− τ)α−1Eα,α (− (t− τ)α) = 1,we may interpret Eα,1 (−tα) and (t− τ)α−1Eα,α (− (t− τ)α), respectivelyas a survival probability up to time t and as the probability density for thebranching at time τ in a branching process Bα. It is a fractional general-ization of an exponential process. This provides a probabilistic sampling ofthe Picard series obtained by iteration of Eq. (9). The solution is thereforeobtained by the expectation of the exit values of the following process:Starting at time zero, a particle lives according to the process Bα. Atthe branching time τ the initial particle dies and two new particles are bornat the dying point. The process continues in the same way with independentevolution of each one of the newborn particles. At time t the solution isobtained as a functional of the n existing particles at time t, namely as theproduct of the initial condition propagated from the point where each oneof the n particles is at time t up to the initial position:u(t, x) = Ex (ϕ1ϕ2 · · ·ϕn) , (11)withϕi =∫dy(i)1 dy(i)2 · · · dy(i)k−1dy(i)k Gβα,α(τ1, x− y(i)1)Gβα,α(τ2, y(i)1 − y(i)2)· · ·· · ·Gβα,α(τk−1, y(i)k−2 − y(i)k−1)Gβα,1(τk, y(i)k−1 − y(i)k)u(0+, y(i)k), (12)with∑kj=1 τj = t, k − 1 being the number of branchings leading to particlei. Notice that the last propagator in (12) is different from the others.Because of the normalization of the probabilities in the process Bα, theprobability of each one of the products in (11) corresponds to the weightof the corresponding term in the Picard series. Therefore the expectationvalue exists whenever the Picard series converges.The solution (11) is not yet a purely stochastic solution because it in-volves both the expectation value over the process Bα and a multiple inte-gration of the initial condition with the propagation kernels Gβα,1 and Gβα,α.To obtain a purely stochastic solution we notice that, for 0 < α ≤ 1, thepropagation kernels satisfy the conditions to be the Green’s functions ofstochastic processes in R (see the Appendix).We denote the processes associated to Gβα,1 (t, x) and Gβα,α (t, x), respec-tively by Πβα,1 and Πβα,α. Therefore the process leading to the solution is asdescribed before with all the particles until the last branching propagating52 F. Cipriano, H. Ouerdiane, R. Vilela Mendesaccording to the process Πβα,α and the last ones (that sample the initial con-dition) propagating by the process Πβα,1. At time t the particle coordinatesx+ ξi are recorded and the solution is given byu(t, x) = Ex(u(0+, x+ ξ1)u(0+, x+ ξ2) · · ·u(0+, x+ ξn)). (13)Eq. (13) is a stochastic solution of (4) and our main result is summarizedas follows:Theorem. The nonlinear fractional partial differential equation (4),with 0 < α ≤ 1, has a stochastic solution given by (13), the coordinatesx+ξi in the arguments of the initial condition obtained from the exit valuesof a propagation and branching process, the branching being ruled by theprocess Bα and the propagation by Πβα,1 for the particles that reach time tand by Πβα,α for all the remaining ones.A sufficient condition for the existence of the solution is∣∣u(0+, x)∣∣ ≤ 1. (14)Remarks:1) The condition |u(0+, x)| ≤ 1 imposes a finite value for all contri-butions to the multiplicative functional. However, the solution may existunder more general conditions, namely when the decreasing value of theprobability of higher order products in (13) compensates the growth of thepowers of the initial condition.2) The stochastic solution may also be constructed by a backwards-in-time stochastic process from time t to time zero. This is obtained byrewriting Eq. (9) asu (t, x) = Eα,1 (−tα)∫ ∞−∞dyF−1Eα,1(−(1 + 12ψθβ (k))tα)Eα,1 (−tα)× (x− y)u (0+, y)+ ∫ t0dττα−1Eα,α (−τα)∫ ∞−∞dyF−1Eα,α(−(1 + 12ψθβ (k))τα)Eα,α (−τα) (x− y)u2 (t− τ, y) , (15)and noticing also thatEα,1 (−tα) +∫ t0dττα−1Eα,α (−τα) = 1.STOCHASTIC SOLUTION OF A KPP-TYPE NONLINEAR . . . 53Then, we obtain the following stochastic construction of the solution:Starting at time t a particle propagates backwards in time according tothe process Πβα,1 if it reaches time zero or according to Πβα,α if it branches attime t− τ . The branching probability is controlled by the process Bα (thatis, the branching probability density is τα−1Eα,α (−τα)). When it branches,two new particles are born which propagate independently and the processis repeated until all surviving particles reach time zero.Appendix: The Green’s functionsand the characterization of the processesThe processes Πβα,1 and Πβα,α:F{Gβα,1 (t, x)}(t, k) =Eα,1(−(1 + 12ψθβ (k))tα)Eα,1 (−tα) , (16)F{Gβα,α (t, x)}(t, k) =Eα,α(−(1 + 12ψθβ (k))tα)Eα,α (−tα) . (17)For a propagation kernel G (t, x) to be the Green function of a stochasticprocess, the following conditions should be satisfied:(i) G (0, x− y) = δ (x− y) or F {G} (0, k) = 1 ∀k;(ii)∫dxG (t, x) = 1 ∀t or F {G} (t, 0) = 1;(iii) G (t, x) should be real and ≥ 0.For the processes Πβα,1 and Πβα,α:(i) F{Gβα,1}(0, k) = Eα,1(0)Eα,1(0) = 1 and F{Gβα,α}(0, k) = Eα,α(0)Eα,α(0) = 1;(ii) F{Gβα,1}(t, 0) = Eα,1(−tα)Eα,1(−tα) = 1 and F{Gβα,α}(t, 0) = Eα,α(−tα)Eα,α(−tα) = 1;(iii) If F {G} (t,−k) = (F {G} (t, k))∗, then G (t, x) is real.Because ψθβ (−k) =(ψθβ (k))∗, it followsEα,1(−(1 +12ψθβ (−k))tα)=(Eα,1(−(1 +12ψθβ (k))tα))∗and54 F. Cipriano, H. Ouerdiane, R. Vilela MendesEα,α(−(1 +12ψθβ (−k))tα)=(Eα,1(−(1 +12ψθβ (k))tα))∗implying that both Gβα,1 (t, x) and Gβα,α (t, x) are real.Finally, for the positivity, one notices that for 0 < α ≤ 1 and ρ ≥ α,Eα,ρ (−x) is a completely monotone function, see [16]. Therefore,Eα,ρ (−x) =∫ ∞0e−rxdF (r) ,with F nondecreasing and bounded.For Gβα,ρ (t, x) (ρ = 1 and ρ = α) one hasGβα,ρ (t, x) =12piEα,ρ (−tα)∫ ∞0dF (r)∫ ∞−∞dke−ikxe−rtα(1+ 12ψθβ(−k))=12piEα,ρ (−tα)∫ ∞0dF (r) e−rtα∫ ∞−∞dke−ikxe−rtα2ψθβ(−k).We recognize the last integral (in k) as the Green’s function of a Levyprocess. Therefore one has an integral in r of positive quantities implyingthat Gβα,1 (t, x) and Gβα,α (t, x) are positive.The process Bα:The decaying probability in time dτ of this process isτα−1Eα,α (−τα) .From ∫ t0τα−1Eα,α (−τα) dτ = 1−Eα,1 (−tα)it follows that Eα,1 (−tα) is the survival probability up to time t. Theprocess Bα is a fractional generalization of the exponential process.References[1] R. Courant, K. Friedrichs and H. Lewy, U¨ber die partiellen Differen-zengleichungen der mathematischen Physik. Mat. Ann. 100 (1928),32-74.[2] R.M. Blumenthal and R.K. Getoor, Markov Processes and PotentialTheory. Academic Press, New York (1968).STOCHASTIC SOLUTION OF A KPP-TYPE NONLINEAR . . . 55[3] R.F. Bass, Diffusions and Elliptic Operators. Springer, New York(1998).[4] C. Graham and S. Me´le´ard, Probabilistic tools and Monte-Carlo aprox-imations for some Boltzmann equations. In: ESAIM Proceedings, vol.10 (Eds. F. Coquel and S. Cordier), Les Ulis (2001), 77-126.[5] Y. LeJan and A.S. Sznitman, Stochastic cascades and 3-dimensionalNavier-Stokes equations. Prob. Theory and Relat. Fields 109 (1997),343-366.[6] E.C. Waymire, Probability & incompressible Navier-Stokes equations:An overview of some recent developments. Prob. Surveys 2 (2005),1-32.[7] R.N. Bhattacharya et al., Majorizing kernels and stochastic cascadeswith applications to incompressible Navier-Stokes equations, Trans.Amer. Math. Soc. 355 (2003), 5003-5040.[8] M. Ossiander, A probabilistic representation of solutions of the in-compressible Navier-Stokes equations in R3. Prob. Theory and Relat.Fields 133 (2005), 267-298.[9] R. Vilela Mendes and F. Cipriano, A stochastic representation forthe Poisson-Vlasov equation. Commun. Nonlinear Science and Num.Simul. 13 (2008), 221-226.[10] E. Floriani, R. Lima and R. Vilela Mendes, Poisson-Vlasov : Stochasticrepresentation and numerical codes. Eur. J. of Physics D 46 (2008),295-302.[11] R. Vilela Mendes, Stochastic processes and the non-perturbative struc-ture of the QCD vacuum. Zeitsch. Phys. C54 (1992), 273-281.[12] A. Kolmogorov, I. Petrovskii and N. Piskunov, E´tude de l’e´quation dela diffusion avec croissance de la quantite´ de la matie`re et son appli-cation a un proble`me biologique. Moscow Univ. Bull. Math. 1 (1937),1-25.[13] F. Mainardi, Y. Luchko and G. Pagnini, The fundamental solution ofthe space-time fractional diffusion equation. Fract. Calc. Appl. Anal.4 (2001), 153-192.[14] H.P. McKean, Application of brownian motion to the equation ofKolmogorov-Petrovskii-Piskunov. Comm. on Pure and Appl. Math. 28(1975), 323-331; 29 (1976), 553-554.56 F. Cipriano, H. Ouerdiane, R. Vilela Mendes[15] A. Kilbas, H. Srivastava and J. Trujillo, Theory and Applications ofFractional Differential Equations. Elsevier B. V., Amsterdam (2006).[16] W.R. Schneider, Completely monotone generalized Mittag-Leﬄerfunctions. Expositiones Mathematicae 14 (1996), 3-16.1 GFM and FCT - Universidade Nova de LisboaComplexo InterdisciplinarAv. Gama Pinto, 2 - 1649 - 003 Lisboa, PORTUGALe-mail: cipriano@cii.fc.ul.pt2 De´partment de Mathe´matiques, Faculte´ des SciencesUniversite´ de Tunis El Manar, Campus Universitaire1060 Tunis, TUNISIAe-mail: Habib.Ouerdiane@fst.rnu.tn3 CMAF, Complexo InterdisciplinarUniversidade de LisboaAv. Gama Pinto, 2 - 1649 - 003 Lisboa, PORTUGALe-mail: vilela@cii.fc.ul.pt& Received: May 25, 2008Centro de Fusa˜o Nuclear - EURATOM/IST AssociationInstituto Superior Te´cnicoAv. Rovisco Pais 1, 1049 - 001 Lisboa, PORTUGAL

Stochastic Solution of a KPP-Type Nonlinear Fractional Differential Equation

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Stochastic Solution of a KPP-Type Nonlinear Fractional Differential Equation

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