Journal of Computational and Applied Mathematics. Tempered stable Lévy motion and transient super-diffusion

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1 Journal of Computational and Applied Mathematics Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: Tempered stable Lévy motion and transient super-diffusion Boris Baeumer a,, Mark M. Meerschaert b a Department of Mathematics & Statistics, University of Otago, Dunedin, New Zealand b Department of Statistics & Probability, Michigan State University, Wells Hall, E. Lansing, MI 4884, United States a r t i c l e i n f o a b s t r a c t Article history: Received 3 September 8 Received in revised form 8 October 9 Keywords: Fractional derivatives Particle tracking Power law Truncated power law The space-fractional diffusion equation models anomalous super-diffusion. Its solutions are transition densities of a stable Lévy motion, representing the accumulation of powerlaw jumps. The tempered stable Lévy motion uses eponential tempering to cool these jumps. A tempered fractional diffusion equation governs the transition densities, which progress from super-diffusive early-time to diffusive late-time behavior. This article provides finite difference and particle tracking methods for solving the tempered fractional diffusion equation with drift. A temporal and spatial second-order Crank Nicolson method is developed, based on a finite difference formula for tempered fractional derivatives. A new eponential rejection method for simulating tempered Lévy stables is presented to facilitate particle tracking codes. 9 Elsevier B.V. All rights reserved. 1. Introduction The diffusion equation t p c p governs the transition densities of a Brownian motion Bt, and its solutions spread at the rate t 1/ for all time. The space-fractional diffusion equation t p c α p for < α < governs the transition densities of a totally skewed α-stable Lévy motion St, and its solutions spread at the super-diffusive rate Sct c 1/α St for any time scale c. The super-diffusive spreading is the result of large power-law jumps, with the probability of a jump longer than r falling off like r α [1 3]. The super-diffusive scaling of a stable Lévy motion cannot be epressed in terms of the second moment, which fails to eist due to the power-law tail of the probability density [4]. Instead one can use fractional moments [5] or quantiles. The totally skewed α-stable Lévy motion St is the scaling limit of a random walk with power-law jumps: c 1/α X X [ct] St in distribution as the time scale c, when the independent jumps all satisfy PX > r Ar α for large >. Here < α < so that the second moment is infinite, and the usual Gaussian central limit theorem does not apply. The random walk with power-law jumps is called a Lévy flight [6]. The order of the fractional derivative in the governing equation t p c α p equals the power-law inde of the jumps. A more general jump variable with PX < r qar α and PX > r 1 qar α leads to a governing equation t p cq α p + 1 qc α p with q 1. A continuous time random walk CTRW imposes a random waiting time T n before the jumps X n, and power-law waiting times PT > t Bt β for < β < 1 lead to a space time fractional governing equation β t p c α p. See [7] for more details. Fractional diffusion equations are important in applications to physics [,3], finance [8], and hydrology [9]. Truncated Lévy flights were proposed by Mantegna and Stanley [1,11] as a modification of the α-stable Lévy motion, since many deem the possibility of arbitrarily long jumps, and the mathematical fact of infinite moments, physically unrealistic. In that model, the largest jumps are simply discarded. Some other modifications to achieve finite second moments were proposed by Sokolov et al. [1], who add a higher-order power-law factor, and Chechkin et al. [13], who Corresponding author. addresses: bbaeumer@maths.otago.ac.nz B. Baeumer, mcubed@stt.msu.edu M.M. Meerschaert /$ see front matter 9 Elsevier B.V. All rights reserved. doi:1.116/j.cam.9.1.7

2 B. Baeumer, M.M. Meerschaert / Journal of Computational and Applied Mathematics add a nonlinear friction term. Tempered Lévy motion takes a different approach, eponentially tempering the probability of large jumps, so that very large jumps are eceedingly unlikely, and all moments eist [14]. Eponential tempering offers technical advantages, since the tempered process remains an infinitely divisible Lévy process whose governing equation can be identified, and whose transition densities can be computed at any scale. Those transition densities solve a tempered fractional diffusion equation, quite similar to the fractional diffusion equation [15]. Like the truncated Lévy flights, they evolve from super-diffusive early-time behavior, to diffusive late-time behavior. This paper provides finite difference and particle tracking methods for solving the tempered fractional diffusion equation. Higher-order methods for fractional diffusion equations are based on the Grünwald finite difference approimation for the fractional derivative [16,17]. This paper develops a second-order Crank Nicolson scheme, using a variant of the Grünwald finite difference formula for tempered fractional derivatives combined with a Richardson etrapolation. Particle tracking solutions for fractional diffusion equations take a Langevin approach, simulating a Markov process whose generator is the adjoint of the fractional derivative operator [18]. For tempered anomalous diffusion, particle tracking can be accomplished using an eponential rejection method for simulating tempered stables. Since the tempered stable process is still an infinitely divisible Lévy process, increments can be simulated for any choice of time step size.. Tempered stable Lévy motion The totally skewed α-stable Lévy motion St for < α <, α 1 has a probability density p, t that solves a fractional diffusion equation: t p, t c α p, t. Take Fourier transforms denoted by ˆf k e ik f d to get d dt ˆpk, t cikα ˆpk, t, recognizing that the fractional derivative α corresponds to multiplication by ikα in Fourier space. The point-source solution ˆpk, t e tcikα inverts to a totally skewed α-stable density with no closed form, ecept in a few eceptional cases [,19]. The constant c > for 1 < α <, and c < for < α < 1. The useful scaling property t 1/α pt 1/α, t p, 1 is evident from the Fourier transform. The process has heavy tails with PSt > r Ar α, so that any moment St n n P, t d of order n > α diverges [4]; hence the need for tempering in some circumstances. The basic idea of eponential tapering is to modify the density function to the form e λ p, t and re-normalize: Lemma..1 on p. 67 in Zolotarev [] shows that the Fourier transform e iu p, t d e tciuα has an analytic etension to Iu <. Setting u k iλ shows that e ik e λ p, t d e tcλ+ikα. For the special case < α < 1, a simpler argument with Laplace transforms appears in Rosiński [14]. Since k yields the total mass, the density e tcλα e λ p, t integrates to one. The Fourier transform is easily computed: e tc[λ+ikα λ α]. Eponential tapering changes the mean: i d dk etc[λ+ikα λ α] k ctαλ α 1. Centering yields ˆp λ k, t e tc[λ+ikα λ α ikαλ α 1]. Invert the Fourier transform to get the mean-zero tempered stable density: p λ, t e λ e ctα 1λα p ctαλ α 1, t. Since d dt ˆp λk, t c[λ + ik α λ α ikαλ α 1 ]ˆp λ k, t, Fourier inversion reveals the tempered fractional diffusion equation t p λ, t c α,λ p λ, t where α,λ f is the inverse Fourier transform of [λ + ik α λ α ikαλ α 1 ]ˆf k. An easy calculation [15] shows that α,λ f e λ α [eλ f ] λ α f αλ α 1 f, 4 which modifies the notation in [15] so that the tempered fractional diffusion equation 3 preserves the center of mass. Everything above carries through without modification for the case α. In this case we can eplicitly compute p, t 4πtc 1/ e /4tc. After substituting into, a little algebra shows that p λ, t p, t. Hence tempering has no effect in the Gaussian case, and a tempered diffusion is eactly the same as a classical diffusion. The case α 1 requires a different form ˆpk, t e tcik lnik to solve 1 with c >. Now the scaling property is tpt + tc ln t, t p, 1. Apply the eponential tempering, normalize and center as before to derive the Fourier representation ˆp λ k, t e tc[λ+ik lnλ+ik λ ln λ ik1+ln λ]. Note that Lemma..1 in Zolotarev [] also applies in this case. Invert to get the mean-zero tempered stable density p λ, t e λ e tcλ p tc1 + ln λ + lntc, t 6 that solves the tempered fractional diffusion equation 3, where α,λ f is the inverse Fourier transform of [λ+ik lnλ+ ik λ ln λ ik1 + ln λ]ˆf k when α

3 44 B. Baeumer, M.M. Meerschaert / Journal of Computational and Applied Mathematics To summarize, a totally positively skewed stable density with inde < α < can be tempered so that moments of all orders eist. The positively skewed tempered stable density is obtained by: i multiplying the stable density by e λ ; ii rescaling by a constant that makes the resulting function integrate to one; and iii shifting by a constant that makes the resulting density function have mean zero. Then we have the following result. Proposition 1. Let St be a totally skewed α-stable Lévy process for < α < with density p, t and Fourier transform E[e ikst ] ˆpk, t { ep [ctik α ] α 1 ep [ctik lnik] α 1. 7 Then the tempered α-stable Lévy process S λ t has density p λ, t { e λ+α 1ctλ α p ctαλ α 1, t α 1 e λ+ctλ p ct1 + ln λ, t α 1. 8 Net we etend Proposition 1 to the general case, to accommodate both positive and negative particle jumps. If St is positively skewed, then St is negatively skewed, with density p, t. For < α <, α 1 the density solves a fractional diffusion equation t p c α p where α corresponds to multiplication by ikα in Fourier space. Tempering multiplies the density by e λ, normalizes to total mass one, and centers, as before. The resulting tempered stable density has Fourier transform e tc[λ ikα λ α +ikαλ α 1] which just replaces ik by ik in the formula 5. It solves the tempered spacefractional diffusion equation t p c α,λ p where α,λ f is the inverse Fourier transform of [λ ikα λ α +ikαλ α 1 ]ˆf k. A similar sign change yields the negatively skewed tempered stable density and governing equation in the case α 1. For the general case, generate two independent positively skewed tempered stable processes S + λ t and S λ t with the same inde α, and set S λ t qs λ t + 1 qs+ λ t where q 1. The probability densities of this process solve the general tempered space-fractional diffusion equation t p cq α,λ p + c1 q α,λ p. Another interesting characterization of the tempered stable process can be stated in terms of its Lévy representation. In short, a Lévy process is the limiting case of a compound Poisson process whose jumps are described by the Lévy measure e.g., see [1, Remark ], and 38. The remainder of this section is devoted to showing that the tempered stable process can also be defined in terms of an eponentially tempered Lévy measure. In the process, we will also show that eponential tempering of the density function can also be used for any Lévy process with positive jumps. Any infinitely divisible process St on R can be characterized by its Fourier transform [ E[e ikst ] ep t aik σ k + e ik 1 + ik ] φd where a R, σ and the Lévy measure φ with φd <, are uniquely determined [4,1]. We call [a, σ, φ] the 1+ Lévy representation. Given an infinitely divisible process St with Lévy representation [,, φ], we define the eponentially tempered Lévy measure φ λ d ep λ φd. It is clear that φ 1+ λ d <, so that there eists another infinitely divisible Lévy process S λ t with Lévy representation [a,, φ λ ] for any a R. Write φ φ + + φ where φ + U φu { : > } is the positive part of the Lévy measure. It is clear from 11 that we can write St S + t + S t where S + t is infinitely divisible with Lévy representation [,, φ + ], S t is infinitely divisible with Lévy representation [,, φ ], and S + t, S t are independent. In order to develop a theory of tempered infinitely divisible laws, it suffices to consider just the positive case. The etension to the signed case is the same as for tempered stable laws. Hence it suffices to consider infinitely divisible laws whose Lévy measure is concentrated on the positive real line. Theorem. Let St be an infinitely divisible process with Lévy representation [a, σ, φ], where φ is supported on R +. Write the Fourier transform E[e ikst ] eptφk and the probability distribution PU, t Pd, t PSt U for Borel U measurable U R. Then Φ has an analytic etension to the lower half plane. Given λ >, define an infinitely divisible process S λ t with Lévy representation [d, σ, φ λ ] where φ λ d e λ φd. The mean E[S λ t] eists for all λ >, and we can choose d R so that E[S λ t]. The resulting tempered infinitely divisible Lévy process S λ t has probability distribution P λ d, t e λ Φ iλt iλφ iλt Pd + iφ iλt, t. 1 Proof. First we show that Φk aik σ k + e ik 1 + ik φd

4 B. Baeumer, M.M. Meerschaert / Journal of Computational and Applied Mathematics has an analytic etension to the lower half plane. For Iz <, let 1 + Ψ z e iz 1 + iz φd. As φ is a Lévy measure, φd <. The rest of the integrand satisfies e iz 1 + iz e iz 1 + iz e iz 1 { z < ep z + < 1 + z / + 1. Hence Ψ is well defined, its comple derivative lim e iz+w 1 + w w 1 lim w w iz + w 1 + e iz e iw 1 + iw + e iz e iw 1 e iz 1 + iz φd 1 + φd 13 i e iz 1 + e iz φd eists by the Dominated Convergence Theorem as 1 e iz e iw 1 + iw + e iz e iw 1 w w < 1 e iz e iw 1 + iw w + e iw 1 e iz w. 15 The second term is bounded by e iz e w < e Iz+ w < M for Iz + w <. For > 1, almost the same bound applies to the first term. For < 1, a Taylor epansion shows that e iz e iw 1 + iw w < M. 16 Note that Ψ k iλ Φk aik σ k e ik λ e ik e ik e λ 1 λ 1 + φd λ φd tends to zero as λ + by another dominated convergence argument, similar to 15 without the 1/w. Hence Ψ z aiz σ z is the analytic etension of Φ. In other words, Φk iλ aik iλ σ k iλ + e ik+λ ik + λ 1 + φd 1 + is the unique etension of Φ. Now e ik 1 + ik e λ φd Φk iλ Φ iλ + aik + σ k ikλσ + bik for b e λ 1 φd R. Let Hd, t denote the probability distribution of the infinitely divisible process with 1+ Lévy representation [a, σ, φ λ ], and denote its Fourier transform by Ĥk, t e ik Hd, t. Then Ĥk, t ep [ t Φk iλ Φ iλ ikλσ + bik ]. As Ĥ is infinitely differentiable, by [, Prop ] all moments are finite. The mean is given by tµ i Ĥ, t k itφ iλ + tλσ bt. Setting d a µ yields an infinitely divisible process S λ t with Lévy representation [d, σ, φ λ ] satisfying E[S λ t] for all t >. Its probability distribution P λ d, t has Fourier transform ˆP λ k, t ep [ t Φk iλ Φ iλ + ikiφ iλ ]. 19

5 44 B. Baeumer, M.M. Meerschaert / Journal of Computational and Applied Mathematics By [3, Thm. III.1], the moment sequence {µ n } with µ n i n n k n ˆp λ, t is positive in the sense of [3, Def. III.9b]. Let qs ˆPλ is, t. Then µ n 1 n dn q ds n and hence by [3, Thm. VI.19.c], the integral qs ˆPλ is, t e s P λ d, t eists for some measure P λ and all Rs > λ. Therefore ˆP λ k iν, t e ik e ν P λ d, t for all k R and ν > λ. As by 19 lim ˆPλ k iν, t [ ep t Φk Φ iλ + ik λiφ iλ ] ν λ + we obtain, using the shift formula for the Fourier transform, that the inverse Fourier transforms are the same as well: e λ P λ d, t e Φ iλt iλφ iλt Pd + iφ iλt, t, which concludes the proof. Remark. If St has a Lebesgue density p, t, so that Pd, t p, td, then it follows immediately from Theorem that the tempered process S λ t has density p λ, t e λ Φ iλt iλφ iλt p + iφ iλt, t. If we take Φk cik α for α 1, or Φk cik logik for α 1, then it follows by an easy computation that 8 holds. This shows that eponentially tempering the stable density is equivalent to eponentially tempering the corresponding Lévy measure. More generally, shows that eponentially tempering the Lévy measure of an infinitely divisible law is equivalent to eponentially tempering the density. 3. Finite difference methods In this section we develop a second-order finite difference method to solve the tempered fractional advection dispersion equation with drift on a bounded interval [ L, R ] with Dirichlet boundary conditions: t u, t v u, t + c α,λ u, t + q, t 1 with u, u, u R, t B R t, and u, t for all < L and t >. We assume v, c and 1 < α, to be consistent with the fractional advection dispersion equation for left-to-right flow [16]. Our approach is similar to [17]. First we develop an implicit Crank Nicolson scheme that is second order in time, but fails to be second order in space, due to the fact that the finite difference approimation of the fractional derivative is only first order. Then we apply Richardson etrapolation to obtain a method that is second order in both variables. We prove second-order consistency, and stability, and conclude with a numerical eample. Recall the fractional binomial formula k α k z k 1 + z α which is valid for any α > and comple z 1. Define Γ k α α w k Γ αγ k + 1 1k k and note that w k can be computed recursively via w 1, w 1 α, w k+1 k α k+1 w k. Our first result establishes a suitable finite difference approimation for the tempered fractional derivative. Recall that L p denotes the class of functions f for which f p d eists, and the Sobolev space W k,p contains the L p functions whose derivatives of order j 1,,..., k are also in L p. Proposition 3. Let f W n+3,1 for some integer n 1. Let p R, h >, λ and 1 < α. Define α h,p f : 1 h α wj e j phλ f j ph 1 e hλ α e phλ j h α f. 3

6 B. Baeumer, M.M. Meerschaert / Journal of Computational and Applied Mathematics Then there eist constants a j independent of h, f, and λ such that α h,pf α,λ f + αλ α 1 f + uniformly in R. n 1 j1 [ a j e λ α+j e λ f ] λ α+j f h j + Oh n 4 Proof. The proof is similar to [17, Proposition 3.1]. Use the fractional binomial formula with z e hλ+ik, along with the fact that e ikaˆf k is the Fourier transform of f a, to see that 3 has Fourier transform φh kˆf k with φ h k 1 w h α j e j phλ+ik e phλ 1 e hλ α j 1 e e phλ+ik hλ+ik α 1 e e phλ hλ h h α λ + ik α ω h λ + ik λ α ω h λ 5 where 1 e hz α ω h z e phz. hz A Taylor series epansion and some elementary estimates imply ω n 1 hλ + ik a j λ + ik j h j Ch n λ + ik n e phλ j for some constants a j and C independent of h. Hence n 1 φ h kˆf k a j λ + ik α+j λ α+j h jˆf k + ˆψk, h j for some function ψ with ˆψk, h Ch n λ + ik α+n + λ α+n e phλ ˆf k. Since f W n+3,1, a standard argument using the Riemann Lebesgue Lemma implies that ˆf k C1 + k n 3. Hence k ψk, h L 1 R and ψ, h C h n e phλ for all R, and then Fourier inversion yields 4 uniformly over R. Remark 4. Note that the approimation 4 has terms involving α+j f. For these to be uniformly bounded, f has to be sufficiently regular, which is the case if f W n+3,1. However, if for eample f β, then α+j f is not bounded near if α + j > β, and 4 will not hold. One should be able to rectify this by using the starting quadrature weights of Lubich [4], but we have not yet attempted this. Net we outline a Crank Nicolson scheme for solving the tempered fractional diffusion equation 1 on [ L, R ] and t [, T]. Define t n n t to be the integration time t n T and h > to be the spatial grid size with i L + i, i,..., N. Define u n i u i, t n, c i c i, v i v i, and q n+1/ i q i, t n+1 + t n /. Let U n i define the numerical approimation to the eact solution u n i. In view of Proposition 3 we will write U n+1 i t U n i v i + c i αλ α 1 δ + c i U δα,λ n+1 i + U n n+1/ i + q i, 6 where δ U n i U n i U n i 1 / and δ α,λ U n i 1 i+1 w α k e k 1λ U n i k+1 Rearranging 6 for U n+1 i k in matri form yields eλ α 1 e λ α U n i. 7 I tau n+1 I + tau n + Q n+1/ t, 8

7 444 B. Baeumer, M.M. Meerschaert / Journal of Computational and Applied Mathematics where U n [U n 1,..., U n N 1 ]T, Q n+1/ [q n+1/ 1, q n+1/,..., q n+1/ N 1 + η N 1e λ B n+1 R + B n R ]T, 9 η i c i α and I is the N 1 N 1 identity matri. The entries of the matri A are collected from 6 to yield η i e λi j w i j+1 j i η i e λ w + c iαλ α 1 + v i j i 1 A i,j η i w1 e λ 1 e λ α c i αλ α 1 + v i j i η i e λ j i + 1 j > i + 1 for i, j 1,..., N 1. The matri A is lower triangular plus entries on the first super-diagonal j i + 1. Note that the boundary condition u N, t n B R t n B n R is incorporated in the forcing term Q, essentially to account for the fact that the super-diagonal term is cut off in the bottom row. Theorem 5. The Crank Nicolson scheme 8 is consistent. Proof. The proof follows immediately from Proposition 3, along with the well-known consistency for the first-order difference approimation δ U n i U n i U n i 1 /. Proposition 6. The fractional Crank Nicolson scheme 8 is unconditionally stable. Proof. The proof is similar to [17, Proposition 3.3]. Note that implies w j e j 1λ e λ 1 e λ α. j i+1 It follows by a straightforward calculation that j A i,j. Since the only negative entries of A are on the diagonal, this implies that A i,i > i+1 j,j i A i,j. Hence the matri is diagonally dominant and using the Gershgorin theorem [5, pp ], all the eigenvalues of the matri A in 3 have negative real parts. Then the spectral mapping theorem shows that the eigenvalues of I ta 1 I + ta have comple absolute value less than 1, so the system is unconditionally stable. Some standard arguments along with Proposition 3 show that the Crank Nicolson scheme is O t +. We can improve the order of approimation by using Richardson etrapolation to obtain second-order convergence; i.e., solve the system twice, once with grid size and again with grid size /. Let U i U i, / U i, be the etrapolated solution on the coarse grid. Then, in view of Proposition 3, the etrapolated scheme is O t +. Eample 7. Consider the tempered fractional diffusion equation with no drift λ α t u, t c e α eλ u, t λ α u, t αλ α 1 u, t + q, t with [, 1], α 1.6, λ, initial condition u, β e λ Γ β + 1, with β.8, diffusion coefficient c α Γ 1 + β α, Γ β + 1 forcing function λ t Γ 1 + β α q, t e Γ β + 1 and boundary condition Bt u1, t e λ t Γ β αλ α α+β Γ β αβλα 1 α+β 1 Γ β β Γ 1 + β α 3

8 B. Baeumer, M.M. Meerschaert / Journal of Computational and Applied Mathematics Table 1 Maimum error behavior for Crank Nicolson CN and etrapolated CNX schemes showing improved second-order convergence. t CN ma error CN error rate CNX ma error CNX error rate 1/1 1/ / 1/ /4 1/ /8 1/ The eact solution is given by u, t β e λ t Γ 1 + β. We solved the equation numerically to time t 1 with and without etrapolation. The etrapolation was done by halving the grid size to / while keeping t to obtain U i, / and then computing U i U i, / U i,. The maimum error is given in Table Monte Carlo simulation Since the tempered α-stable Lévy process S λ t is infinitely divisible, it can be approimated by a random walk at any given mesh t. In this section, we provide a general method to generate tempered stable random variables. More generally, the method can be used to simulate any eponentially tempered random variable. Given a non-negative random variable X with density f, define the eponentially tempered density f λ e λ f / e λu f u du. Take Y eponentially distributed with mean λ 1, independent of X. Generate X i, Y i independent and identically distributed with X, Y, and let N min{n : X n Y n }. Then X N has the eponentially tempered density function f λ : To see this, let q PX Y PX yλe λy dy and compute PX N PX N, N n where n1 PX n, X n Y n PX n 1 > Y n 1 PX 1 > Y 1 n1 PX n, X n Y n 1 q n 1 n1 PX n, X n Y n q 1 PX n, X n Y n PX, X yλe λy dy PX yλe λy dy + PX e λ 3 and hence the density of X N is d d PX N q 1 d [ ] PX yλe λy dy + PX e λ d q 1 [ PX λe λ + f e λ PX λe λ] 33 which reduces to f λ, since q f ye λy dy via integration by parts. If St is stable with inde α < 1 and its density has Fourier transform pk, t e tcikα then St, and the method of [6,19] can be used to simulate St Z: Take U uniform on [ π/, π/], W eponential with mean 1, and set Z c t 1/α sin αu + π cosu 1/α cosu αu + π 1 α/α. 34 W The tempered stable random variable S λ t can be simulated by the rejection method: Draw Z, and Y eponential with mean λ 1. If Y < Z, reject and draw again; otherwise, set S λ t Z + ctαλ α 1. To simulate the entire path of the process, write t n t and note that S λ t n k1 [S λk t S λ k 1 t], a random walk with all n summands independent and identically distributed with S λ t. 31

9 446 B. Baeumer, M.M. Meerschaert / Journal of Computational and Applied Mathematics When α > 1, the stable density p, t with Fourier transform ˆpk, t e tcikα is positive for all < < and all t > ; i.e., the support is all of R. The eponentially tempered density p λ, t e tcλα e λ p, t is still integrable because p, t faster than any eponential function as [, Theorem.5. and Eq ]. To simulate these random variables, we adapt the procedure for α < 1 and use the following theorem. Recall that for two random variables X, Z with cumulative distribution functions F PX, Gz PZ z, the Kolmogorov Smirnov distance is defined by dx, Z sup{ F G : R}. Theorem 8. Let X be a random variable with density f such that the integral I e λ f d eists. Define the eponentially tempered density f λ e λ f /I. Let Y be eponentially distributed with mean λ 1, independent of X. Generate X i, Y i independent and identically distributed with X, Y, and for a let N a min{n : X n Y n a}. Then the density of X Na converges in L 1 to f λ as a, and hence dx Na, X in the Kolmogorov Smirnov distance. Proof. Set q PX Y a and compute F a PX Na q 1 PX, X Y a. Note that F a q 1 PX for < a, and, similar to 3, for a we have +a F a q 1 PX y aλe λy dy + PX e λ+a. Then the density of X Na is f a q 1 f for < a and, by the same argument as 33, f a q 1 f e λ+a for a. Note that q PX y aλe λy dy e λa PX yλe λy dy a f y dy + e λa f ye λy dy. a a Let h a e λa a f y dy and g a a e λy f y dy. Then q e λa I g a + h a, h a g a and the L 1 distance Finally we note that f λ f a d dx Na, X sup R sup R sup R a e λ f f I q d + e λ f f e λ+a a I q d e λ f eλa f I I g a + h d + e λ f f e λ a a I I g a + h d a g a 1 + I g a 1 3g a. I g a + h a I g a + h a I I g a + h a a F λ F a which completes the proof. f λ y dy f λ y f a y dy f λ y f a y dy f a y dy 3g a I g a + h a 35 Note that if PX a 1, then obviously g a and X Na has the eponentially tempered function f λ as its density. To simulate S λ t for α > 1, choose a > so that PSt < a is small, draw Z using Z ct 1/α sin αu π cosu 1/α cosu αu π + π α 1 α/α 36 W and Y eponential with mean λ 1. If Z > Y a, reject and draw again; otherwise, set S λ t Z +ctαλ α 1. Note that a larger a increases accuracy of the simulation method, however a larger a will result in more rejections, increasing computing time. For α 1, use the same method and set S λ t Z + ct1 + ln λ where π W cos U Z ct ln π π ct + + U tanu ln π + U. 37

10 B. Baeumer, M.M. Meerschaert / Journal of Computational and Applied Mathematics λ.1 λ λ.1 λ Fig. 1. Tempered stable Lévy motion S λ t with α 1.. A random walk simulation of the particle path can be accomplished as in the case α < 1, by simulating and adding independent and identically distributed random jumps S λ t. Fig. 1 illustrates a random walk simulation of S λ t with α 1., c 1, t 1, and t T 1, for four different values of the tempering parameter λ. The simulation produced no Z <.5, and the chosen shift a 4 lead to few rejects [4 out of 1, for λ.1; 35 for λ.1; 3 for λ.1; and 5 for λ.1]. In this case, the random variable S1 is stable with inde α, mean zero, skewness +1, and scale σ cosπα/ 1/α.3578 in the parameterization of Samorodnitsky and Taqqu [19]. The Kolmogorov Smirnov bound 35 is less than 1 14 zero to machine precision which can be verified by numerical integration using the fast efficient codes of Nolan [7] for the stable density. Observe that as λ gets smaller, the anomalous large jumps in Fig. 1 increase, leading to super-diffusion. Particle tracking codes can be accomplished by simulating a large number of particles, and generating a histogram of the results at different time points. The particle tracking method can be etended to variable coefficient equations with complicated geometry and boundary conditions. An alternative simulation method uses the Poisson approimation for an infinitely divisible Lévy process. It follows from results in Cartea and del Castillo-Negrete [15] that the density p λ, t of S λ t has Fourier transform [ ] ˆp λ k, t ep c t e ik 1 + ike λ α α 1 d where c cα 1/Γ α. Note that this corrects the constant in [15, Eq. 1]. Given ε > small, we estimate ˆp λ k, t by [ ] [ ] ep c t e ik 1 + ike λ α α 1 d ep m ε c tˆfε k 1 + ikµ ε 38 ε where m ε ε e λ α α 1 d, f ε m 1 ε e λ α α 1 is a probability density, and µ ε ε f ε d is its mean. Recognizing that the last line in 38 is the Fourier transform of a mean-centered compound Poisson [1], we can write Nεt S λ t J i µ ε i1 where J i are independent random variables with density f ε, and N ε t is a Poisson process with rate m ε c. The approimation becomes eact as ε. To simulate J i, use the eponential rejection method for positive random variables X developed in this section, with PX > /ε α, so that X ε/u 1/α with U uniform on [, 1]. To simulate the Poisson process, let W i lnu i /m ε c, T n W W n, and N ε t ma{n : T n t}. Note that this approimation is actually a continuous time random walk [,3] with eponentially tempered power-law jumps. Setting λ yields a well-known approimation to the anomalous diffusion process St. In the special case α < 1, a more refined version of this simulation idea was developed in a recent paper of Cohen and Rosiński [8]. Their method reproduces the eact sample path of the process up to small jumps, using a series representation based on a similar Poissonian representation. Acknowledgements We would like to thank M. Kovaćs and the anonymous reviewers for the fruitful discussions and suggestions on improving the manuscript. M.M. Meerschaert was partially supported by NSF grants DMS-8336 and EAR This paper was completed while B. Baeumer was on sabbatical leave at the Department of Statistics & Probability, Michigan State University.

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