Continuous time random walk models for fractional space-time diffusion equations
aa r X i v : . [ m a t h . P R ] S e p Continuous time random walk models forfractional space-time diffusion equations
Sabir Umarov
University of New Haven, Department of Mathematics,300 Boston Post Road, West Haven, CT 06516, USA
Abstract
In this paper continuous time random walk models approximating fractional space-time diffusion processes are studied. Stochastic processes associated with the con-sidered equations represent time-changed processes, where the time-change processis a L´evy’s stable subordinator with the stability index β ∈ (0 , . In the parer theconvergence of constructed CTRWs to time-changed processes associated with the cor-responding fractional diffusion equations are proved using a new analytic method.
Keywords : random walk, stochastic process, time-changed process, fractional order differen-tial equation, pseudo-differential operator, L´evy process, stable subordinator.
By definition, a continuous time random walk (CTRW) is a random walk subordinated to arenewal process. More precisely this means that CTRW comprises of two i.i.d. sequences ofrandom variables (vectors), X n expressing jumps, and T n expressing waiting times betweensuccessive jumps. CTRW model was introduced by Montrol and Weiss [1] in 1965. CTRWprocesses have broad applications in various fields (see, e.g. [2, 3]) and have been extensivelyinvestigated recent years by many authors; see [2]-[8] and references therein. If the sequences X n and T n are independent, then STRW is called decoupled. We would like to single outtwo papers [5, 6], contents of which are close to the present paper, and which discuss acontinuous time random walk approximation of the stochastic process associated with thefractional order diffusion equation of the form D β ∗ u ( t, x ) = 12 ∂ u ( t, x ) ∂x , t > , x ∈ R . (1)Here D β ∗ is the fractional derivative of order β ∈ (0 ,
1) in the sense of Caputo-Djrbashian: D β ∗ g ( t ) = 1Γ(1 − β ) Z t g ′ ( τ )( t − τ ) β dτ. (2)1ne can always assume that the process starts from the origin, which in terms of u ( t, x ) , iswritten in the form u (0 , x ) = δ ( x ) , (3)where δ is the Dirac delta function concentrated at 0 . These papers suggest two distinctdiscretizations of equation (1). Both discretizations lead to the unified form D β ∗ u nj ≈ a ( τ ) u n +1 j − c u nj − n X m =2 c m u n +1 − mj − γ n u j ! , n = 1 , , . . . , (4)where u nj = u ( t n , x j ) , t n = nτ, τ > , x j = hj = ( hj , . . . , hj d ) , and u j = 1 , if j = 0 =(0 , . . . , , and u j = 0 , if j = 0 . In [5] the Gr¨unwald-Letnikov approximation is used fordiscretization of D β ∗ u ( t, x ) , and parameters a ( τ ) , γ n , and c k are defined as a ( τ ) = a ( τ, β ) = 1 τ β , c k = c k ( β ) = ( − k (cid:18) βk (cid:19) , k = 1 , , . . . , (5) γ n = γ n ( β ) = n X k =0 c k , n = 1 , , . . . . (6)Paper [6] uses a special quadrature for the discretization of D β ∗ u ( t, x ) with parameters a ( τ ) = 1 τ β Γ(2 − β ) , γ m = ( m + 1) (1 − β ) − m (1 − β ) , m = 0 , , . . . , n, (7) c = 1 , c k = γ k − − γ k , k = 1 , . . . , n, (8)for an arbitrary fixed n. For the right hand side of (1) the second order symmetric fi-nite difference approximation is used. These discretisations lead to an explicit scheme u n +1 j = F ( u j , . . . , u nj ) , where F is a linear function of its arguments. The positivenessand conservativeness conditions, as well as the stability and convergence issues are studiedin these papers using numerical methods.In our paper we develop an analytic method for convergence of the CTRW to the stochas-tic process associated with the distributed order diffusion equation D β ∗ u ( t, x ) = Z D α u ( t, x ) dρ ( α ) , t > , x ∈ R d , with initial condition (3). Here D α , < α < , is a hypersingular integral operator definedin the form D α f ( x ) = b ( α ) Z R ny ∆ y f ( x ) | y | n + α dy, (9)with ∆ y , the second order centered finite difference in the y direction and b ( α ) is the nor-malizing constant b ( α ) = α Γ( α )Γ( n + α ) sin απ − α π n/ , and ρ is a finite measure with support in the interval [0 , . In this paper we will use only theGr¨unwald-Letnikov approximation for D β ∗ . The method developed in this paper is a modifi-cation of the method used in [9, 10] and provides a rigorous justification of the convergenceof constructed CTRWs to their associated limit stochastic processes.2
Preliminaries and auxiliaries
Let ρ be a finite measure with the support supp ρ ⊆ [0 , . Denote by SS the class of d -dimensional ( F t )-semimartingales Z t , Z = 0 , whose characteristic function is given by E (cid:2) e iξZ t (cid:3) = exp n − t Z | ξ | α dρ ( α ) o , ξ ∈ R d . (10)If f Z t ( x ) , x ∈ R d , is the density function of the process Z t , then equation (10) can beexpressed as the Fourier transform E (cid:2) e iξZ t (cid:3) = F [ f Z t ]( ξ ) . The class SS obviously containsL´evy’s symmetric α -stable processes and all mixtures of their finitely many independentrepresentatives. For the process Z t ∈ SS corresponding to a finite measure ρ , we use thenotation Z t = X ρt to indicate this correspondence.Suppose X ρt ∈ R d is a stochastic process obtained by mixing of independent L´evy’s SαS -processes with a mixing measure ρ, supp ρ ⊂ (0 , . Then its associated FPK equation hasthe form ∂u ( t, x ) ∂t = Z D α u ( t, x ) dρ ( α ) , t > , x ∈ R d , (11)where D α is defined in (9). Indeed, it is seen from (10) that the L´evy symbol of X ρt equalsΨ( ξ ) = − Z | ξ | α dρ. (12)On the other hand, due to the known relationship F [ D α ϕ ]( ξ ) = −| ξ | F [ ϕ ]( ξ ) (see, e.g. [11]),the Fourier transform of the right hand side of (11) is F h Z D α ϕ ( x ) dρ ( α ) i = Z F [ D α ϕ ]( ξ ) dρ ( α )= (cid:16) − Z | ξ | α dρ (cid:17) F [ ϕ ]( ξ ) = Ψ( ξ ) F [ ϕ ]( ξ ) , ξ ∈ R d . Hence, the symbol of the pseudo-differential operator on the right hand side of (11) coincideswith Ψ( ξ ) . This means that the FPK equation associated with X ρt is given by equation (11).Since X ρ = 0 the function u ( t, x ) in (11) satisfies the initial condition u (0 , x ) = δ ( x ) . Usingthe Fourier transform technique, one can verify that the solution of (11) satisfying the latterinitial condition can be expresses in the form G ρ ( t, x ) = 1(2 π ) n Z R n e t Ψ( ξ ) − ixξ dξ. (13)We note also that a pseudo-differential operator A ( D ) and its symbol σ A ( ξ ) are connectedthrough the relation [12] σ A ( ξ ) = e − ixξ A ( D ) e ixξ valid for all x. In particular, if x = 0 : σ A ( ξ ) = (cid:0) A ( D ) e ixξ (cid:1) | x =0 , ξ ∈ R d . (14)3et X ρt ∈ SS and Y t = X ρW t be the time-changed process, where W t is the inverse toa β -stable subordinator (see [13]). Then the density of Y t solves the fractional FPK typeequation [14] D β ∗ u ( t, x ) = Z D α u ( t, x ) dρ, t > , x ∈ R d , (15)with a finite measure ρ, supp ρ ⊂ (0 , , and initial condition (3). The unique solution ofequation (15) satisfying initial condition (3) is given by [15] u ( t, x ) = E β (Ψ( D ) t β ) δ ( x ) = 1(2 π ) d Z R d E β [Ψ( ξ ) t β ] e − ixξ dξ, where E β ( z ) is the Mittag-Leffler function [16]. Thus, the Fourier transform of u ( t, x ) inthe variable x is F [ u ( t, · )]( ξ ) = E β (Ψ( ξ ) t β ) . Moreover, it follows from the propertyies of theMittag-Leffler function that the Fourier-Laplace transform of u ( t, x ) is L [ F [ u ]( ξ )]( s ) = s β − s β + ( − Ψ( ξ )) , s > , ξ ∈ R d . We will use the Gr¨unwald-Letnikov approximation for discretization of the Caputo-Djrbashianfractional derivative in equation (15). By definition, the backward Gr¨unwald-Letnikov frac-tional derivative of order β of a function, defined on an interval ( a, b ) , is a D βt f ( t ) = lim h → τ β ⌊ t − aτ ⌋ X m =0 ( − m (cid:18) βm (cid:19) f ( t − mτ ) . (16)The Gr¨unwald-Letnikov fractional derivative (16) can be used for approximation of theCaputo-Djrbashian derivative (2). Namely, taking τ = ( t − a ) /n and t k = a + kτ, k = 0 , . . . , n, in (16), one has the approximation [17] D β ∗ f ( t n ) ≈ τ β n X m =0 ( − m (cid:18) βm (cid:19)(cid:16) f ( t n − m ) − f ( a ) (cid:17) . (17)with the order of accuracy O ( τ ) . Moreover, if a = −∞ , then the finite sum in (16) becomesan infinite series, that is ∞ D βt f ( t ) = lim τ → τ β ∞ X m =0 ( − m (cid:18) βm (cid:19) f ( t − mτ ) , (18)convergent for functions satisfying the asymptotic behavior f ( t ) = O ( | t | − (1+ β + ε ) ) , t → −∞ , (19)for some ε > . It is also known [11] that in the class of such functions ∞ D βt f ( t ) coincideswith the Liuoville-Weyl backward fractional derivative of order β, defined as −∞ D βt f ( t ) = 1Γ(1 − β ) ddt Z t −∞ f ( τ )( t − τ ) dτ, (0 < β < . f ( t ) = e st , s > , defined on ( −∞ , b ] , < b < ∞ , obviouslysatisfies condition (19). It can be readely veryfied that −∞ D βt e st = s β e st . Hence, taking t = 0 , one has (cid:16) ∞ D βt e st (cid:17) | t =0 = (cid:16) −∞ D βt e st (cid:17) | t =0 = s β . (20)This example will be used in in the proof of the main result. Lemma 2.1
Let < α < and p k = b ( α ) | k | − ( d + α ) , k = 0 , − b ( α ) X m ∈ Z d \{ } | m | d + α , k = 0 , (21) where Z d is the d -dimensional integer lattice. Then the characteristic function ˆ p h ( ξ ) = X k ∈ Z d p k e ikhξ , of the sequence { p k } k ∈ Z d , converges to −| ξ | α / as h → . Proof.
Let ˆ p h ( ξ ) be the characteristic function of p k defined in (21), that isˆ p h ( ξ ) = X k ∈ Z d p k e ihkξ = − b ( α ) X m ∈ Z d \{ } | m | d + α + b ( α ) X m ∈ Z d \{ } | m | d + α e imhξ = − b ( α ) X m ∈ Z d \{ } | m | d + α (1 − e imhξ ) . Further, one can easily verify that X = k ∈ Z d − e ikξh | k | d + α = X = k ∈ Z d − e − ikξh | k | d + α . Due to the definition of the symmetric second finite difference of the function e ixξ at theorigin, this impliesˆ p h ( ξ ) = − b ( α ) X = k ∈ Z d − e ikξh | k | d + α = − b ( α ) 12 X = k ∈ Z d − e ikξh + e − ikξh | k | d + α = b ( α ) 12 X = k ∈ Z d (∆ kh e ixξ ) | x =0 | k | d + α . Now letting h → , due to definition (9) of D α and relation (14), we havelim h → ˆ p h ( ξ ) = 12 b ( α ) Z R d (∆ y e ixξ ) x =0 | y | d + α dy = 12 ( D α e ixξ ) | x =0 = − | ξ | α , as desired. 5 orollary 2.2 Let d k := Q k ( h ) | k | d , k = 0 , − X k =0 Q k ( h ) | k | d , k = 0 , (22) where Q k ( h ) = 2 Z b ( α ) dρ ( α ) h α | k | α , k = 0 . (23) Then the characteristic function ˆ d h ( ξ ) of the sequence { d k } k ∈ Z d , converges to Ψ( ξ ) as h → . Proof
We have ˆ d ( hξ ) = − X k =0 Q k ( h ) | k | d + X k =0 Q k ( h ) | k | d e ikhξ = X k =0 Q k ( h ) | k | d ( e ikhξ −
1) = 12 X k =0 Q k ( h ) | k | d (cid:0) e ikhξ − e ikhξ (cid:1) = 12 X k =0 Q k ( h ) | k | d (cid:0) ∆ kh e ixξ (cid:1) | x =0 = Z b ( α ) X k =0 ∆ kh e ixξ | kh | d +2 h d ! | x =0 dρ ( α )= Z ˆ p h ( ξ ) dρ ( α ) , where ˆ p h ( ξ ) is the charactersitic function of the sequence p k in equation (21). Letting h → , due to Lebesgue’s dominated convergence theorem and Lemma 2.1, we obtainlim h → ˆ d ( hξ ) = − Z | ξ | dρ ( α ) = Ψ( ξ ) . As is known [14, 18, 19], driving processes of stochastic differential equation associatedwith time-fractional Fokker-Planck-Kolmogorov (FPK) equations appear to be time-changesof basic processes like Brownian motion, L´evy process, fractional Brownian motion, etc.Donsker’s theorem states that Brownian motion is the limit in the weak topology of a scaledsum of a sequence of independent and identically distributed (i.i.d.) random variables { X j } , with X ∈ L ( P ). This fact is important from the approximation point of view since anapproximation of the basic driving process B t yields, under some conditions, an approxima-tion of other processes X t driven by B t . Natural approximants of time-changed processes B W , L W , etc., where W is the inverse to a stable subordinator, are CTRWs. A decoupledCTRW is defined by two independent sequences of random variables/vectors: one represent-ing the sizes of jumps, the other representing waiting times between successive jumps. Moreprecisely, let Y , Y , . . . , Y n , . . . , ( Y i ∈ R d ) ,
6e a sequence of i.i.d. random vectors, and let T , T , . . . , T n , . . . , ( T i ∈ R + )be an i.i.d. sequence of positive real-valued random variables. Then S n = Y + · · · + Y n is the position after n jumps, and t n = T + · · · + T n is the time of the n th jump. Assume that S = 0 and t = 0. The stochastic process X t = S N t = N t X i =1 Y i , where N t = max { n ≥ t n ≤ t } , is called a continuous time random walk .Since Y t = X ρW t is a non-Markovian process, its approximating random walk also can nothave independent members. Therefore, transition probabilities split into two different setsof probabilities:1. non-Markovian transition probabilities, which express a long non-Markovian memoryof past; and2. Markovian transition probabilities, which express transition from positions at the pre-vious time instant.Suppose that non-Markovian transition probabilities are given by c ℓ = ( − ℓ +1 (cid:18) βℓ (cid:19) = (cid:12)(cid:12)(cid:12)(cid:18) βℓ (cid:19)(cid:12)(cid:12)(cid:12) , ℓ = 1 , . . . , n,γ n = n X ℓ =0 ( − ℓ (cid:18) βℓ (cid:19) , (24)and Markovian transition probabilities { p k } k ∈ Z n are given by p k = ( c − τ β Q ( h ) , if k = 0; τ β Q k ( h ) | k | d , if k = 0, (25)where Q k ( h ) , k = 0 , is defined in (23), and Q ( h ) = P k =0 Q k ( h ) | k | − n . Then CTRW, approx-imating the stochastic process Y t , can be interpreted in the following sense: the probability q n +1 j of the walker being at a site x j = jh, j ∈ Z d , at a time t n +1 is q n +1 j = γ n q j + n − X ℓ =1 c n − ℓ +1 q ℓj + (cid:16) c − τ β Q ( h ) (cid:17) q nj + X k =0 p k q nj − k . (26)7o construct CTRW (26) one needs to discretize (15). Namely, for the Caputo fractionalderivative on the left-hand-side of (15) we use the backward Gr¨unwald-Letnikov discretization(17) in the form: D β ∗ u nj = D βt u nj ≈ n +1 X m =0 ( − m (cid:18) βm (cid:19) u n +1 − mj − u j τ β (27)where u nj = u ( t n , x j ) , n = 0 , , . . . , j ∈ Z d , x j ∈ h Z d , and t n = nτ, τ > . Using notations(24) and rearranging terms, equation (27) can be expressed in the form D β ∗ u nj ≈ τ β u n +1 j − c u nj − n +1 X m =2 c m u n +1 − mj − γ n u j ! . (28)For the right hand side of (15) we use the discretizationΨ( D x ) u nj ≈ X k ∈ Z d d k u nj − k , d k := Q k ( h ) | k | d , k = 0 , − X k =0 Q k ( h ) | k | d , k = 0 , (29)where Q k ( h ) is defined in (23). Setting the discretizations for the time and space-fractionalderivatives in (28) and (29) equal to each other, we get1 τ β u n +1 j − c u nj − n X m =2 c m u n +1 − mj − γ n u j ! = X k ∈ Z d d k u nj − k . (30)Rearranging terms and solving for u n +1 j in equation (30), the following recursion equation isconstructed, reconstituting CTRW (26): u n +1 j = γ n u j + n X m =2 c m u n +1 − mj + X k ∈ Z d q k u nj − k , (31) q k = τ β d k = 2 Z (cid:18) τ β h α (cid:19) dρ ( α ) | k | d + α , k = 0 c − X k =0 q k , k = 0 . By construction, u j = 1 if j = 0 = (0 , . . . , , and u j = 0 otherwise.The update u n +1 j in equation (31) is determined by Markovian contributions (those valuesof u at time t = t n ) and non-Markovian contributions (those values of u at times t = { t , t , . . . , t n − } ). The order of the time fractional derivative β determines the effect thatthe non-Markovian transition probabilities ( γ n and c , . . . , c m ) has on u n +1 j . This effect canbe measured by sum of all of the transition probabilities in equation (31): γ n + n X m =2 c m ! + X k ∈ Z d q k = 1 . X k ∈ Z d q k = ( c − q ) + X k =0 q k = c and γ n + n X m =2 c m = 1 − c . As a result, when β = 1 one has c = 1, c = · · · = c n = γ n = 0 , and hence, equation (31)simply reduces to u n +1 j = X k ∈ Z d p k u nj − k , j ∈ Z d , n = 0 , , . . . . with the transition probabilities p k = ( − τ P m =0 Q m ( h ) | m | d , if k = 0; τ Q k ( h ) | k | d . if k = 0, (32) Theorem 3.1
Let < β ≤ . Fix t > and let h > , τ = t/n. Let Y j ∈ Z d , j ≥ , be identically distributed random vectors with the non-Markovian and Markovian transitionprobabilities defined in (24) and in (25), respectively. Assume that τ ≤ (cid:16) βQ ( h ) (cid:17) β . (33) Then the sequence of random vectors S n = hY + ... + hY n , converges as n → ∞ in law to X t = Y W t whose probability density function is the solution to equation (15) with the initialcondition u (0 , x ) = δ ( x ) . Proof.
Let ˆ u n ( ξ ) be the characteristic function of the discrete sequence u nj for a fixed n = 0 , , . . . . Then equation (31), in terms of characteristic functions, takes the formˆ u n +1 ( ξ ) = γ n + n X m =2 c m ˆ u n +1 − m ( ξ ) + ˆ q ( ξ )ˆ u n ( ξ ) , (34)since ˆ u ( ξ ) = 1 . Further, let ˆ U τ ( s, ξ ) be the discrete Laplace transform of ˆ u n +1 ( ξ ) , namelyˆ U τ ( s, ξ ) = τ ∞ X n =0 ˆ u n +1 ( ξ ) e − st n , s > . Then multiplying both sides of (34) by τ e − nτs and summing over the index n, one obtainsˆ U τ ( s, ξ ) = γ τ ( s ) + τ ∞ X n =0 n +1 X m =2 c m ˆ u n +1 − m ( ξ ) ! e − nτs + ˆ q ( ξ ) τ ∞ X n =0 ˆ u n ( ξ ) e − snτ = γ τ ( s ) − τ ∞ X n =0 n +1 X m =1 ( − m (cid:18) βm (cid:19) ˆ u n +1 − m ( ξ ) ! e − nτs + ˆ d ( ξ ) τ β ∞ X n =0 ˆ u n ( ξ ) e − snτ , (35)9here γ τ ( s ) = τ ∞ X n =0 γ n e − snτ = τ ∞ X n =0 n +1 X m =0 ( − m (cid:18) βm (cid:19) e − snτ . Changing the order of summation one can show that γ τ ( s ) = e sτ ∞ X n =0 τ e − snτ ! ∞ X m =0 ( − m (cid:18) βm (cid:19) e − smτ . In order to prove the theorem we need to show that ˆ U τ ( s, hξ ) converges as h → τ → L [ E β (Ψ( ξ ) t β )]( s ) = s β − s β + (cid:0) − Ψ( ξ ) (cid:1) , the Laplace transform of the Mittag-Leffler function E β ( x ) composed by Ψ( ξ ) t β . Indeed,this convergence implies the convergence ˆ u n ( hξ ) → E β (Ψ( ξ ) t β ) , as n → ∞ , uniformly for all ξ ∈ K , where K is an arbitrary compact in R d . In turn, the latter convergence is equivalentto the convergence in law of the sequence S n to the process Y W t . To show the convergenceˆ U τ ( s, hξ ) → L [ E β (Ψ( ξ ) t β )]( s ) , we notice that τ ∞ X n =0 ˆ u n ( ξ ) e − snτ = τ + e − sτ ˆ U τ ( s, ξ ) , (36)and changing the order of summation τ ∞ X n =0 n +1 X m =1 ( − m (cid:18) βm (cid:19) ˆ u n +1 − m ( ξ ) ! e − nτs = − τ β + (cid:16) τ e sτ + ˆ U τ ( s, ξ ) (cid:17) ∞ X n =0 ( − n (cid:18) βn (cid:19) e − snτ − ! . (37)It follows from equations (35)-(37) thatˆ U τ ( s, ξ ) = e sτ I τ ( β, s ) ∞ X n =0 τ e − snτ − τ ! + τ − β ( τ β ˆ d ( ξ ) + β + e sτ ) I τ ( β, s ) − ˆ d ( ξ ) e − sτ (38)where I τ ( β, s ) = 1 τ β ∞ X n =0 ( − n (cid:18) βn (cid:19) e − snτ . Further, the following limits hold:lim τ → τ β ∞ X n =0 ( − n (cid:18) βn (cid:19) e − snτ = (cid:16) −∞ D βt e st (cid:17) | t =0 = s β , (39)10im τ → ∞ X n =0 τ e − snτ − τ ! = s − , (40)lim h → ˆ d ( hξ ) = Ψ( ξ ) , (41)lim τ → τ − β ( τ β ˆ d ( ξ ) + β + e sτ ) = 0 , (42)The relation (39) follows from the definition (18) with f ( t ) = e st , s > , and the relation(20). The relations (40) and (42) can be easily verified by direct calculation. The relation(41) is proved in Corollary to Lemma 2.1. Now taking into account the relations (39)-(42)it follows from (38) that lim h → ˆ U τ ( s, hξ ) = s β − s β − Ψ( ξ ) , proving the theorem. Theorem 3.1 extends to the case when the left hand side of equation (15) is a time dis-tributed fractional order differential operator with a mixing measure µ whose support satis-fies supp µ ⊆ [0 ,
1] : D µ u ( t, x ) = Z D β ∗ u ( t, x ) dµ ( β ) = Ψ( D ) u ( t, x ) , t > , x ∈ R d , (43)where Ψ( D ) is a pseudo-differential operator with the symbol Ψ( ξ ) defined in (12). In thiscase for the left hand side of (43) we again have a discretization of the form (4). Namely,we have D µ u nj ≈ a ( τ ) u n +1 j − c ∗ u nj − n X m =2 c ∗ m u n +1 − mj − γ ∗ n u j ! , where a ( τ ) = Z a ( τ, β ) dµ ( β ) , c ∗ k = Z c k ( β ) dµ ( β ) , k = 1 , . . . , n, (44) γ ∗ n = Z γ n ( β ) dµ ( β ) , n = 1 , , . . . . (45)In equations (44) and (45) the integrands a ( τ, β ) , c k ( β ) and γ n ( β ) are defined in (5),(6) or(7),(8) depending on whether the Gr¨unwald-Letnikov or quadrature approximation in paper[6] is used for discretization of D β ∗ u ( t, x ) in (43). The detailed analysis of the correspondingCTRW including simulation models will be presented in a separate paper.We also note that condition (33) takes the form τ ≤ (cid:16) − − β Γ(2 − β ) Q ( h ) (cid:17) β
11f the non-Markovian probabilities are selected as in paper [6, 10]. This condition as wellas (33) generalize the well-known Lax’s stability condition τ ≤ h / β = 1 . In this case Q ( h ) reduces simply to Q ( h ) = 2 /h . Finally, in the particular case β = 1 , Theorem 3.1 reduces to the following theorem,which provides a random walk approximation of stochastic processes X ρt ∈ SS . Theorem 4.1
Let X j ∈ h Z d , j ≥ , be i.i.d. random vectors with the probability massfunction p k = P ( X = k ) defined in (32) with some τ > , h > . Assume that σ ( τ, h ) := 2 τ X m =0 Q m ( h ) | m | n ≤ . Then the sequence of random vectors S N = X + ... + X N , converges in law as N → ∞ to X ρt ∈ SS , whose probability density function is G ρ ( t, x ) defined in (13), that is the solutionto equation (11) with the initial condition u (0 , x ) = δ ( x ) . This theorem in the particular case dρ ( α ) = a ( α ) dα, where a ( · ) is a positive continuousfunction on the interval [0 , , is proved in [9, 10]. References [1] Montroll E W and Weiss G H 1965 J. Math. Phys. IMS Lect. Notes -Monograph Series: High DimensionalProbability Fractional Integrals and Derivatives:Theory and Applications (Gordon and Breach Science Publishers)1212] H¨ormander L 1985
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