Approximation of Levy-Feller Diffusion by Random Walk

Zeitschrift für Analysis und ihre Anwendungen Journal for Analysis and its Applications Volume 18 (1999), No, 2, 231-246 Approximation of Levy-Feller Diffusion by Random Walk R. Gorenflo and F. Mainardi Dedicated to Prof. L. von Wolfersdorf on occasion of his 65th birthday Abstract. After an outline of W. Feller's inversion of the (later so called) Feller potential operators and the presentation of the semigroups thus generated, we interpret the two-level difference scheme resulting from the Grünwald-Letnikov discretization of fractional derivatives as a random walk model discrete in space and time. We show that by properly scaled transition to vanishing space and time steps this model converges to the continuous Markov process that we view as a generalized diffusion process. By re-interpretation of the proof we get a discrete probability distribution that lies in the domain of attraction of the corresponding stable Levy distribution. By letting only the time-step tend to zero we get a random walk model discrete in space but continuous in time. Finally, we present a random walk model for the timeparametrized Cauchy probability density. Keywords: Stable probability distributions, Riesz-Feller potentials, pseudo-differential equations, Markov processes, random walks AMS subject clissification: 26 A 33, 44 A 20, 45K 05, 60 E 07, 60 J 15, 60 J 60 1. Introduction Let 0 < a <2 and II a if0<a<1 i 12— a fl<a<2 (9 real) and denote by p 0 (x; 9) for x E R the stable probability density whose characteristic function (Fourier transform) is j30 (ic; 9) exp ( - ( E R) (1.2) 1, [17], (see, e.g., [ 4 [19] for the general theory of stable probability distributions). In particular we recommend [4], Feller's parametrization being close to ours. For a generic function f on R we denote by f its Fourier transform Ie' K x f (x) dx (kER) (1.3) R. Gorenflo: Free Univ. of Berlin, Dept. Math. & Comp. Sci., Arnimallee 2-6, D-14195'Berlin e-mail: gorenflo@math.fu-berlin.de F. Mainardi: Univ. of Bologna, Dept. Phys., Via Irnerio 46, 1-40126 Bologna, Italy e-mail: mainardi@bo.infn.it ISSN 0232-2064 / $ 2.50 © Heldermann Verlag Berlin 232 R. Gorenflo and F. Mainardi +00 and we then have, in the case of f If(")I I dic < 00, +00 1(x) = '-2ir j °(x e-'--!(.)d,, E R). -00 For i > 0 we rescale Pa by the similarity variable x 9 a (x,t;9) tpa (xt*;9) to obtain (x ER, i >0). (1.4) This function g 0 ( . , t;O) again is a stable probability density, and by interpreting x as space and t as time variable we have in g a description of a Markov process that can be considered as a generalized diffusion process. In fact, we have in x2 92(x,t;0)=t 2exp__ the classical Gauss process and in 91(x,t;0) 7r X2 +t2' the Cauchy process. For a few other pairs (a, 0) leading to elementary or well-investigated special functions, see [19]. A general representation of all stable probability densities in terms of Fox H functions has been only recently achieved (see [181). The Fourier transform of g,, being = exp ( - ( K E R) (1.5) we recognize g a (x,t;O) as the fundamental solution (Green function for the Cauchy problem) of the pseudo-differential equation ôu(x, t) =Du(x,t) (xER,t>0) (1.6) where the pseudo-differential operator .D has the symbol values u(x,0)=f(x) (xER,JEL 1 (R)) ,c)! For initial (1.7) we then have as solution to (1.6) u(x,t) =Jg a (x -,t;O)f(e)d and for all t > 0 then u(-, t) E C 00 flL 1 (R) and f u (x, t) dx =Jf(x)dx. ( 1.8) Approximation of Levy-Feller Diffusion 233 William Feller in his pioneering paper [ 3 ] has shown that the pseudo-differential operator can be viewed as the operator inverse to the Feller potential operator (the name "Feller potential" is used in [161) which is a linear combination of two Weyl integrals. Honouring both Levy and Feller for their essential contributions [11], [12) and [3] we call the process described by (1.6) Levy-Feller diffusion. We now give, in our notation, a formal account of the essentials of Feller's theory (for more details see [8]). With the Weyl integrals J (x - e)°()de J ( - x)()d I (Iç o)(x) = (Io)(x) = and (for 0 < a _ - C + (a; 9) < r(a) S 1 (xER) (1.9) J I 2 but a 54 1) the coefficients - 9)) sin ((a + 9)) =_________ S_ c_=c_ = sin(a7r) ' sin(air) (1.10) and (by passing to the limit a = 2) (1.11) the Feller potentials are given as (I° )(x) = c_(a, 9)(Ijp)(x) + c+(a, O)(Iço)(x). (1.12) Note that in accordance with [16] we omit the singular case a = 1. Feller [3] has shown the operator I to possess the semigroup property for 0<a,/3<1 with a+8<1, II=I° ' and so analytic continuation to negative exponents can he justified to obtain the operator D — I = — {c+(a,9)I;°+c_(a,9)I:'} (1.13) for 0 < a 2 but a 54 1, the parameter 9 restricted as in (1.1), with (see [16)) - dx d2 2-o {± I0 if0<a< 1 ifl<a<2. (1.14) From [3], equating - 2a to Feller's parameter S, we take the symbol of the pseudodifferential operator D as =— Jr J ' e 'In particular, we have D = but D 234 R. Gorenflo and F. Mainardi For the rest of this paper we always keep in mind the distinction of the following two cases: . . (a) 0 < & < 1 and 1 9 1 :5 a. (b) 1<a <2 and 101< 2— a Henceforth, for ease of notation, we shall omit the arguments of the coefficients c = c+(a,0) and c_ = c_(a,0). We have (>0 in the case (a) 1<0 in the case (b) and COS c++c— — cos Or { air > 0 in the case (a) <0 in the case (b). (1.16) The reader is asked not to worry about the foregoing purely formal description of Feller's considerations. It will merely serve us as a motivation for constructing a difference scheme via the Grünwald-Letnikov discretization of fractional derivatives, a difference scheme which by interpretation as a random walk model will be shown to converge (in a sense to be specified in Section 3). 2. Random walks, discrete in space and time In this section we define a random variable Y assuming only integers as values, its probability distribution depending on three parameters a, 0 and p. By aid of this random variable we define a random walk on an equidistant grid { jh lj E Z} with a space-step h > 0. We show that after introduction of a time-step r > 0 this random walk admits an interpretation as an explicit difference scheme for the Cauchy problem (1.6) - (1.7), namely for R,t >0) &(2.1) ôu(x,t) =D(x,t) (x u(x,0) - AX). In the next section we shall show that the probability distribution of the discrete random variable Y belongs to the domain of attraction of the Levy distribution with the parameters a and 9, proceeding in a way which simultaneously proves "convergence" of the random walk (if r = - 0) to the corresponding Levy-Feller diffusion characterized by (1.4). Let Y be a random variable assuming its values in Z, P(Y = k) = Pk for k E Z, with probabilities Pk defined as follows. With a parameter ,u, restricted by { air 2 in the case (a) 97r cos -11 Icos air I -_ --i in the case (b) a cos-- (2.2) Approximation of Levy- Feller Diffusion 235 put in the case (a) P0 = 1 - p(c+ + c_) Pk =(_1)i.tc+() (2.3) = (— l)zc_ () P-k (k e N) and in the case (b) pol+,ia(c+*c_) P1 = — [+() +c_] = _[_() +c+] . . - (2.4) Ce Pk =(_1)c+(kl), P-k (_1)c_(k ) (k > 2). One sees that all Pk ^! 0, and by rearrangement it turns out that Pk = 1 - (c + c_) kEZ (-1) () = 1 — 0. j=0 Remark 2.1. It is worthwhile here to observe the fact which will also be useful in Section 3 that for all a > 0 the series > k _o(- 1 ) k ( ak ) z k for (1 - z) a converges absolutely and uniforrnly 'on the closed unit disk I i 1, due to the asymptotics ()I '-.r(a + 'for k - , valid for non-integer a > 0. This asymptotics can be deduced by use of the reflection formula for the gamma function and Stirling's asymptotics. We obtain a random walk on the grid random variables S {jh l j E Z} starting at the point 0, by defining = hY + hI'2 +... + hY,, (n E N) (2.5) with the Y2 as independent identically distributed random variables, all having the same probability distribution as the random variable Y. Let us write our random walk in an alternative way. Discretizing the space variable x and the time variable t by grid points x 3 = Jh and instants t,, = nr, with h > 0, T> 0, j E 7Z, n E No and denoting by y, (1,,) the probability of sojourn of the random walker in point x 2 at instant I,,, the recursion S n+1 = S,, + hY,,+ 1 (following from (2.5)) means 7J(i,,+ i ) = pkyj_k(t,,) (j E 7L, fl E N0 ), (2.6) kEZ and the random walker starting, at point xo = 0 means y(0) = 1 and y3 (0) = 0. for j. 0. However, in the recursion scheme (2.6) it is legitimate to use a more general initial sojourn probability distribution {y 3 (0)j E Z}. There is yet another possible interpretation of (2.6), namely as a redistribution scheme of an extensive quantity (e.g. mass, charge, or may be probability), y,(t,,) being imagined as a clump of this extensive quantity, sitting in point x j at instant I,,. Then (2.6) is a conservative and non-negativity 236 R. Gorenflo and F. Mainardi preserving redistribution scheme. In fact, from all immediately for all n E N that = JEZ y(0) ^! 0 and E kCZ Pk if jEZ all y,(t) ^! 0 Pk 1 it follows <c jEZ if all y(0) > 0. Such redistribution schemes have been shown to be useful for discretization of diffusion processes modelled by second order linear parabolic differential equations (see, e.g., [6], [7], [9]) as they discretely imitate essential properties of the continuous process. To come nearer to the Cauchy problem (2.1) we relate the time step step h by the scaling relation T= T to the space (2.7) and remark that the y(t) are then intended as approximations to r+4 J zj —. which, if u( . , i) is continuous, is also hu(x,, ta). It is again a matter of rearrangement to show that (2.6) is equivalent to the explicit difference scheme T = hDo°Yj(in) (2.8) where (in analogy to (1.13)) hD = — {c h I + c hL° } with the GrünwaldLetnikov discretization (see [16]) of the fractional derivatives in the form h hIYj = h (_1)k()yjk in the case (a) >I(_i)k()yj±lk in the case (b). ko (2.9) Notice the shift of index in the case (b) which among other things has the effect that in the special case a = 2 (the classical diffusion equation) we obtain the standard symmetric three-point difference scheme. For more details and discussions see [8]. Instead of trying to work out a convergence proof for the difference scheme (2.8), thereby using the Lax-Richtmyer theory of consistency, stability and convergence (in effect the Lax equivalence theorem, see [5] or [141) we prefer to present in the next section a proof in the true spirit of random walks. We leave the numerical analysis aspect to a forthcoming paper. . Approximation of Levy-Feller Diffusion 237 3. Convergence and domain of attraction We will show that for fixed t ni- > 0 the discrete distribution of the sojourn probabilities y(t) (j e Z) with initial condition y(0) = ^jo (Kronecker symbol) converges completely to the probability distribution with density g a (x,t;9) tp(xt;9) (x E JR) (3.1) as n —* +00. Let us remind that this probability distribution has the characteristic function t; 9) = fga(x, t; 9) e"' dx = exp ( — t II e'' ). (3.2) To avoid confusion of language one meets in probability theory let us agree to use the terminology adopted in [10]. From this source we take Definitions 3.1 - 3.4, Remark 3.1 and Theorem 3.1. Definition 3.1. Let (F) be a sequence of uniformly bounded, non-decreasing right-continuous functions defined on R. We say that F converges weakly to a bounded non-decreasing right-continuous function F on JR if F(x) — F(s) at all continuity points of F. In this case we write Fn +F. Definition 3.2. Let (F) be as in Definition 3.1. Then (F) is said to converge completely to F if (i) F-4F and (ii) F(00) — F(00) as n — 00. In this case we write F-+F. Theorem 3.1 (Continuity theorem). Let (F) be a sequence of probability distribution functions, and let (pn) be the sequence of the corresponding characteristic functions, () =fe c ' dF(r) (te E JR). Then (F)converges completely to a probability distribution function F if and only if — a(ic) for all c E JR as n —* co, where i() is continuous at c = 0. In this case the limit function cp is the characteristic function of the limit distribution function F, () =fe' dF(x) (r. E R). Definition 3.3. In the cases where the functions F and F are probability distribution functions such that F-+F, let X and X be random variables corresponding to F and F, respectively. Then we say that X,, converges in law to X. Definition 3.4. Let (X) be a sequence of independent identically distributed random variables with common probability distribution function F. Suppose there 238 R. Gorenflo and F. Mainardi exist sequences (an) and (bn) of constants, with b > 0, such that the sequence of sums b Xk— a converges in law to some random variable with probability distribution function G. Then we say that F is attracted to G. The set of all probability distribution functions attracted to G is called the domain of attraction of the distribution function G. Remark 3.1. A stable probability distribution is characterized by having a domain of attraction. Let us now state, with the notations of Sections 1 and 2, our Theorem 3.2. Let the independent identically distributed random variables Y 1 , Y2, Y3 ,... have the common probability distribution function F of the random variable Y with P(Y = k) = Pk, k E Z, and Pk given by (2.3), (2.4), respectively. Let X(t) with > 0 be the random variable with probability density ga (x, t; 9) and let G(, t ) 9) be the corresponding distribution function, Ga(x,t;9) =J g,t;9)d. Then F is attracted by Ga( . ,t ) 9), indeed: forn — * no the distribution function of the random variable (3.3) converges completely to G 0 ( . , t; 9), the distribution function of the random variable X(t). Proof. Using the scaling relation (2.7), namely r = ph o , and the substitution = nr of time, we get h = (t/(n))*, (3.4) and comparing (3.3) and (2.5) we see that X,, = S,, the random variable taking values in the grid {jhlj E 7L} at the fixed instant t, = nr = t. In view of Theorem 3.1 and the fact that (ic, t; 9) is continuous at ic = 0, it only remains to prove that the characteristic function of the sojourn probabilities yj(tn), namely the function t; h) = y,(t) e'', with tn = t, (3.5) jEz tends for all icE R (as h — * 0) to ü0(,c,t;9) = exp (- t i c e' (5 ""). Let us calculate (c, t; h) for ease of notation via the generating functions and jEZ (3.6) JEZ of the transition probabilities and the sojourn probabilities. The series in (3.6) converge absolutely and uniformly on the periphery J zJ -* 1 of the unit circle, representing there a continuous function, and due to the fact that the random walk occurs on the grid {jhi, E 7L} change to. characteristic functions j3(c) and (ic,tn) is accomplished via Approximation of Levy-Feller Diffusion 239 = e 1 '' (ic E R). Using the binomial series for (1 - z), absolutely convergent on Izi = 1 if a > 0 (see Remark 2.1), we readily verify the identities p{c+(1 — (z) - 1 - z) + c_(1 — z_} in the case (a) (3.7) in the case (b). {c+z(1 - z) + c_z(1 — z')} From the discrete convolution (2.6) we deduce (z, t) = (z, 0) ((z)), and the special initial condition y(0) = jO for j E Z gives (z,0) 1, hence (z,t) In view of (3.4), (3.8) and the fixation t ((z))' as n — . —* = ((z))'. = t,, = nr we have to show that, with z = e"", )) exp (- More clearly, using (2.7) and t t; h) (3.8) = t; 9) = t, = nr, we have to show that the function = ((e"")) (# E IR) (3.9) has the property lirn Q(K, t; h) = (ic, t; 9). (3.10) Let us first treat the case (a): 0< a <land 1 9 1 a. Then z11cIhsign x )°+ = 1 — 1L{c+(1 — c_(1 — ic =IKI sign , and zklhsIgn We see that j3(e'°") = 1, whereas we can get the result for ic < 0 by complex conjugation of that for , > 0. So, for notational ease, we treat in detail the case r, > 0. In this case j5(e1ch) = 1 - {c+(1 —e sch) + c_(1 — e")} (3.11) and for small h by Taylor (1 — e") and = (— ikh + O(h2))° = (_0 0 ( K h) a (1 + O(h))' = e(,ch)a + o(ha+l) 0 — etc)0 = e' f (kh) + O(hc). Inserting this into (3.7) we find = 1— h{c^ei!P +c_e!t} + O(h') By use of (1.10) for c and c_ and the complex omple' represntation of sin forward calculation yields for (fixed) r. > 0 =1- h''e' + O(h'') = 1 - ,1 1 k 1 l 'e'i a straight- + O(h''), 240 R. Gorenflo and F. Mainardi for t < 0 by complex conjugation = 1 - ,2 1, Ih 0 e_ t + 0(h") So, finally, for all ic e ji(e") = 1— + 0(h') (3.12) and by (3.9) log (k, t; h) = = _tI,cIc i(s +0(h"+1)} PC) 12' + 0(h), hence, as desired, (3.10). In the case (b): 1 <a < 2 and 1 9 1 2 - a, we have by (3.7) = 1 - i{c+e'(1 - + c_e"(1 - and in comparison to the case (a) we have because of et P( h = 1 + 0(h) within {. . .} the additional asymptotic term 0(h)(1 - + 0(h)(1 - = 0(h'), hence again (3.12) for all c E R, and again we arrive at (3.10) U It is instructive to take a look at the very special case a = 2 and 9 = 0 (the classical diffusion equation Uj = u 1 ). In this case Az) = 1 + 12{ - 2 + = 1 +- e and one finds ((e')) }2 = 1— 4psin2 ?Ch 2 - exp(— ti 2 ) as h - 0. 4. A random walk model, discrete in space, continuous in time Consider the difference scheme (2.8) which is equivalent to the redistribution scheme (or random walk model) (2.6) with the coefficients given by (2.3) or (2.4), respectively. By sending the parameter i - 0 (letting the time step T tend to 0) we obtain an infinite system of ordinary differential equations = h' ;y1(i) y (0 ) given (j J e Z) (4.1) Approximation of Levy-Feller Diffusion 241 describing a time-continuous redistribution scheme over the grid {yh j E Z} in time i 2 0 of the form y(t) = ;qkYj_k(t) (4.2) (j E Z). kEZ Interpreting y3 (t) as a clump of an extensive quantity sitting in point x = J at instant we have, for (4.2) to describe such a redistribution scheme, the balancing conditions qo <0 q (4.3) 20 (0 k E Z) } = 0. qk (4.4) kEZ In analogy to our redistribution scheme (2.6) of Section 2 system (4.2) also is conservative and non-negativity preserving. In fact, it can be shown (we leave this as an exercise to the reader) that system (4.2) under conditions (4.3) - (4.4) is uniquely solvable if i I(°)I <oo (4.5) )Ez and that then E,EZ I y ( t )I < 00 for all t > 0. It can further be shown that then y(t)=0, hence JEZ y(t)Ey(0) jEZ JEZ for all t > 0. If furthermore yj( 0) > 0 (j E Z), then y1 (t) 2 0 (j E Z) for all I > 0. The interpretation of (4.2) with (4.3) and (4.4) as a redistribution scheme means: lqoly j (t) is the rate of outflow from the point x 3 = jh being transferred to other points, and this must equal the sum of the rates q k y(t), received by the points Xj+k (k 54 0). Using in (4.1) again the Grünwald-Letnikov discretization (2.9) we find the following. In the case (a) 0 < a < 1 and 101 <a: cos- ) q o = -h - "(c+ + c_) = -h° cos qk q-k (4.6) = h_a(_1)Ic+() (k EN) = h_(_1)k+1c_() (kEN). j In the case (b) 1 <a<2 and 101<2— a: -a q o = h(c+ + c_)cx = -ah qi = -h qk = (_1)kh_aC^(k q-k = (_l)kh_a c_(k [+ () cos 'I IcosI c+] I + c_], q_j = -h°[_ () + ) (k 22) ) (k 22). 1 . J (4.7) 242 R. Gorenflo and F. Mainardi By playing again with infinite sums of binomial coefficients it is readily verified that (4.3) and (4.4) are fulfilled. For solving system (4.2) with initial values y (0 ) (j E Z) with >, I(° )I < _ and given by (4.6) (4.7), we apply the method of generating functions. The series qk and q(z)=r >qkzk (z, t) = >Yk(t)Z k kEZ kEZ (4.8) converge absolutely and uniformly on the periphery Izi = 1 of the unit circle, and system (4.2) is then (with Izi = 1) equivalent to ô(z,t) = at y(t)zi = JEZ (kYi_k(t) jEZ kEZ , hence we have the z-parameterized ordinary differential equation problem O(z, t) at = (z)(z, t) (t 20, 1.1 1)I (z, O) = >Yk(0)zk. kEZ The solution is (z, t) = (z,0) et( z ) , or simply (z, t) = t(z) . ( 4.9) in the special case y(0) b j o , for j E Z which means (z, 0) By inspection (using the binomial series) we see that - q(z) = + c_(1 - I — h {c + (1 - z) I — h{c+z'(1 - z)' Changing to characteristic functions via of Section 3 for small h + c_z(1 _z_1)a} in the case (b). z = e' (n E lim h— O (K t; h) = R), we take from our calculations + 0(h). = Then with (4.9) we get for (K,t;h) in' the case (a) z)} := (e",t) exp ( - tIicIe''" in analogy to (3.10) the limit relation sc)!t) = ( #c, t; 9) (c e R). We have ' interpreted (4.2) as a time-continuous redistribution scheme. We can interpret it probabilistically as a random walk model discrete in space (over the grid { j hlj E 7L}), but continuous in time. At any instant i of time the random walker can jump to another grid point. After arriving at a point Xm he will remain sitting there for a random time interval whose length is exponentially distributed. More precisely: Approximation of Levy-Feller Diffusion 243 when we know that at instant t he is sitting at point Xm, then the conditional sojourn probabilities for sitting at points x j are = *5mj (j E 7L) and (4.2) gives by re-conditioning the equation = — lqoIim(t), for the time interval [t*, i+i) of sojourn at Xm. Its solution is iim(t) = 1 - _ lqol(t_i) (t > 1), from which we deduce that the time i the wanderer remains sitting at any pbint Xm is exponentially distributed with parameter Iqo I. Hence, the random walker, after arriving at point Xm sits there for a random time interval of length t and then jumps to another point x 3 in instant t = t + i. The conditional probability of jumping to the point x3 (with j 54 ni) is then given as 1fff. For general information on time-continuous discrete Markov processes we refer the reader to [15). It should finally be remarkedthat the conditional density iim(t) (t > t) can also be obtained in the limit of r - 0 from the conditional geometric probability distribution relevant in the random walk model (2.6) with the transition probabilities of (2.3) - (2.4) and the scaling condition (2.7). We can now state a theorem analogous to Theorem 3.2, namely •Theorem 4.1. Let a random walker start in point 0 at instant t = 0 and jump over the grid points j h (j E Z) with h > 0, the probabilities y 3 (t) of sojourn in point jh at instant t 2 0 evolving according to (4.2) with y2 (0) = jO•Then for fixed t > o the distribution function G 0 ( . , t; 9; h) given as Gc,(x, t; 9; h) = y(t) converges kh<x - completely to Ga(• ,t;9) as h — * 0. - 5. A random walk model for the Cauchy process For completeness we present a random walk model, discrete in space and time, for the omitted case c = 1, namely for the Cauchy process (ci = 1 and 9 = 0). This model cannot be obtained via the Grünwald-Letnikov approach, neither directly nor by a passage to the limit ci 1. We have, with the notations of Sections 1 - 3 the process 1 t 22 =t -1 P( (x E R,t>0) with the Cauchy probability density 7r(x 2 + 1) (xER). The corresponding characteristic functions are Pi(-; 0) = e N and t; 0) Let Y be a random variable assuming its values in Z with P(Y = k) = defined as follows. With a parameter it restricted to 0 < it put = 1-r Pk It 7rIkI(jkI+1) 54 (0kZ). Pk for k E 7L (5.1) 244 R. Gorenflo and F. Mainardi Then all Pk ^: 0 and >kEZPk = 1. Indeed, C-0 00 =1. k(k+l) — — Ti) Proceeding as in Sections 2 and 3 we produce a random walk on the grid with h > 0, letting the walker start in point 0 at instant 0. We define S =h Yi+hY2+...+h y {jhlj E Z}, (nN) where the Yk are independent identically distributed random variables all having their distribution common with the random variable Y. By allowing the walker to jump in instants t,. = nr (r > 0, n E N) and using S, as the random variable assuming values in the grid { .yhj E 7Z} we have a random walk. We relate the steps h and r of space and time by r = p h and again define generating functions ji(z) = > pi z3 and = (z, t,,) iEZ yj(tn)z (5.2) JEZ with y j (tn) as probability of sojourn in point x 3 = jh at instant t, = nr. Then (z, t) = ((z))72, and in view of our aim is to show that for all tc E R and t > 0 the limit relation e_ t II as h — 0 (5.3) holds. This limit relation then implies that the distribution of the random variable —* Y lies in the domain of attraction of the Cauchy distribution, more precisely that the distribution of the random variable x=L{Y1+Y2+...+Y} Lfl converges completely to the Cauchy distribution C 1 (., t; 0) with Gi(x,t;0)=f Let us prove (5.3). We observe that the series E jc z pj z 3 converges absolutely and uniformly on the periphery 1z = 1 of the unit circle, hence represents there a continuous function so that the following calculations are legitimate. In fact, 00 zk k(k+ 1) = 1— (1— z 1 )log(1 — z) for Izi 1. We tacitly take the limit 1 for z = 1. Then (z) = 1— {(i — z')log(l — z) + (1— z)log(1 — Passing to the characteristic functionvia z = e = 1— {(1 — cos(kh))logll — e"I (K z ' )}. E R) we get — sin(ich) arctan 1 Using lim.+, arctan u = ±we obtain for h — + 0 the asymptotics = 1— which implies (5.3). 22 7r (5.4) {l K I h 2 + o (Iic l h )} =1 - ikI h + o(h) } Approximation of Levy- Feller Diffusion 245 6. Concluding remarks It is instructive to observe that in view of Theorem 3.1 our proof of Theorem 3.2 can be re-interpreted as a proof of existence of Levy's stable distributions (for a 5A 1). In fact, assuming to be ignorant of these we can find them as limiting distributions by sending n - in (3.3). And gratis (the discrete probabilities being all non-negative cannot become negative in the limit) we get that the limiting densities are everywhere non-negative, for all values of the parameter a between 0 and 2, with omission of the value 1. For this we actually need neither the theory of the inversion of the Feller potentials nor the method of positive-definite functions. Thus we have an alternative way of solving a problem that surmounted Cauchy's capabilities [2] who had considered the functions exp(— IKI°) as candidates of cosine transforms of probability densities but could only prove them to have this property in the special cases a = land a = 2. Levy in [11], (12] introduced the whole scale of stable densities, Bochner in [1] has given an elegant proof for the full range 0 < a 2 that the inverse Fourier transforms of the functions exp(_I,cI o ) are non-negative, hence probability densities. He used the theory of positive-definite functions that we can avoid. A well readable account of Bochner's method can be looked up in [13]. Acknowledgements. We are grateful to the Italian Consiglio Nazionale delle Ricerche and to the Research Commission of Free University of Berlin for supporting joint efforts of our research groups in Berlin and Bologna. This paper is one of the fruits of this collaboration. We appreciate the careful work of the referees, in particular the constructive-critical comments of one of them. References [1] Bochner, S.: Stable laws of probability and completely monotone functions. Duke Math. J. 3 (1937), 726 - 728. [2] Cauchy, A.: Calcul des probabilités. Stir les résultats moyens d'observations de mime nature, et sur les résultats les plus probables. Stir la probabilitédes erreurs qui effectent des résultats moyens d'observations de mime nature. Comptes rendus 37 (1853), 198 206 and 264 - 272. Reprinted in: Oeuvres Completes d'Augustin Cauchy, Ser. I, Tome 12, pp. 94 - 114. Paris: Gauthier-Villars 1900. (3] Feller, W.: On a generalization of Marcel Riesz' potentials and the seinigroups generated by them. Meddelanden Lunds Universitets Matematiska Seminarium (Comm. S5m. Mathém. Universitéde Lund), Tome suppl. dédiéa M. Riesz. Lund 1952, pp. 73 - 81. [4] Feller, W.: An Introduction to Probability Theory and its Applications, Vol. 2, 2nd ed. New York: Wiley 1971 (1st ed. 1966). [5] Goldstein, J. A.: Semsgroups of Linear Operators and Applications. Oxford - New York: Oxford Univ. Press 1985. [6] Gorenflo, R.: Nichtnegativitiits- und substanzerhaltende Differenzenschemata ftir lineare Diffusionsgleichungen. Numer, Math. 14 (1970), 448 - 467. [7] Gorenflo, R.: Conservative difference schemes for diffusion problems. In: Intern. Ser. Nurner. Math.: Vol. 39). Basel: Birkhäuser-Verlag 1978, pp. 101 - 124. 246 R. Gorenflo and F. Mainardi [8] Gorenflo, R. and F. Mainardi: Random walk models for space-fractional diffusion processes. Fract. Cal. AppI. Anal. 1(1998), 167 - 191. 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