Fully discrete random walks for space–time fractional diffusion equations

Signal Processing 83 (2003) 2411 – 2420 www.elsevier.com/locate/sigpro

Fully discrete random walks for space–time fractional di#usion equations Rudolf Goren'oa;∗ , Alessandro Vivolib a Department

of Mathematics and Informatics, Free University of Berlin, Arnimallee 3, D-14195 Berlin, Germany of Physics, University of Bologna, Via Irnerio 46, I-40126 Bologna, Italy

b Department

Received 6 March 2003

Abstract For space–time fractional di#usion equations a theory of discrete-space discrete-time random walks, analogous to the theory of continuous-time random walks, is presented. The essential assumption is that the probabilities for waiting times and jump-widths behave asymptotically like powers with negative exponents related to the orders of the fractional derivatives. Properly scaled passage to the di#usion limit then leads to the space–time fractional di#usion equation. Illustrating examples are given, numerical results and plots of simulations are displayed. ? 2003 Elsevier B.V. All rights reserved. MSC: 26A33; 44A10; 45K05; 47G30; 60G18; 60G50; 60G51; 60J60 Keywords: Discrete random walk; Master equation; Space–time fractional di#usion; Fractional calculus; Asymptotic power laws

1. The continuous-time random walk and the fractional diusion equation The concept of continuous time random walk (CTRW) was developed in Statistical Mechanics by Montroll and Weiss in [23], see e.g. [27]. The CTRW can be understood by simply considering a random walk subordinated to a time renewal process, see e.g. [3], as pointed out in [1,14,19,26]. The CTRW arises by a sequence of independent identically distributed (iid) positive random waiting times T1 ; T2 ; : : : ; each having probability density ∗

Corresponding author. E-mail addresses: goren'[email protected] (R. Goren'o), [email protected] (A. Vivoli). URL: http://www.fracalmo.org

function (pdf) (t), t ¿ 0, and a sequence of iid random jumps X1 ; X2 ; X3 ; : : : in R, each having a pdf w(x), x ∈ R. Setting t0 = 0, tn = T1 + T2 + · · · Tn for n ∈ N, 0 ¡ t1 ¡ t2 ¡ · · · ; the wandering particle starts at point x = 0 in instant t = 0 and makes a jump of length Xn in instant tn , so that its position is x = 0 for 0 6 t ¡ T1 = t1 , and Sn = X1 + X2 + · · · Xn

for tn 6 t ¡ tn+1 :

An often (also by us) required assumption is that the waiting time distribution and the jump width distribution are independent of each other. It is well known that this stochastic process is Markovian if and only if the waiting time pdf is of the form (t) = me−mt with some positive constant m (compound Poisson process), see e.g. [5]. Then, by natural probabilistic arguments we arrive at the

0165-1684/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0165-1684(03)00193-2

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R. Goren2o, A. Vivoli / Signal Processing 83 (2003) 2411 – 2420

master equation for the probability density function p(x; t) of the particle being in point x at instant t, see [18,22,25],
t (t − t  ) p(x; t) = (x)(t) + 
×

0

+∞

−∞

 w(x − x )p(x ; t  ) d x dt  ; (1.1)

in which (x) denotes the Dirac function,  ∞ generalized and, for abbreviation, (t)= t (t  ) dt  , is the probability that at instant t the particle is still sitting in its starting position x = 0. Clearly, (1.1) satisGes the initial condition p(x; 0) = (x). 1 ˆ Throughout this paper we will denote by f() and g(s) ˜ the transforms of Fourier and Laplace, respectively, of suKciently well-behaved (generalized) functions f(x) and g(t) according to
+∞ ˆ eix f(x) d x;  ∈ R; F{f(x); } = f() =
L{g(t); s} = g(s) ˜ =

−∞

+∞

0

e−st g(t) dt;

s ¿ s0

and consistently, we will have for the Dirac delta funcˆ ˜ ≡ 1. Note that for our purposes we tion () ≡ (s) agree to take s real in the Laplace transform. Then in the Fourier–Laplace domain the master equation (1.1) appears as ˜ p(; ˜ˆ s) = (s) + ˜ (s)w() ˆ p(; ˜ˆ s);

=

u(x; 0+ ) = (x);

0 ¡  6 2; 0 ¡  6 1;

x ∈ R; t ¿ 0:

(1.4)

In view of the particular initial condition, the solution of this Cauchy problem is referred to as the fundamental solution or the Green function. The fractional Riesz derivative x R is deGned as the pseudo-di#erential operator with symbol −|| . This means that for a suKciently well-behaved (generalized) function f(x) we have 2 ˆ F{x R f(x); } = −|| f();

 ∈ R:

(1.5)

The symbol of the Riesz fractional derivative is nothing but the logarithm of the characteristic function of the generic symmetric stable (in the LPevy sense) probability density, see [4,5,24]. Noting −|| = −(2 )=2 , we recognize that 

d2 xR = − − d x2 

=2 :

(1.6)

(1.3)

˜ − s−1 f(0+ ); L{t D∗ f(t); s} = s f(s)

˜ (s) 1 − w() ˆ

Our aim is to derive from the master equation (1.1), by properly rescaling the waiting times and the jump widths and passing to the di#usion limit, the space–time fractional di9usion equation. This is a 1

= x R u(x; t);

In other words: the Riesz derivative is a symmetric fractional generalization of the second derivative. The Caputo fractional derivative in time provides a fractional generalization of the Grst derivative through the following rule in the Laplace transform domain

˜ (s)

1 1 − ˜ (s) : ˜ (s) s 1 − w() ˆ

 t D∗ u(x; t)

(1.2)

whence p(; ˜ˆ s) =

partial pseudo-di#erential equation, which is obtained from the standard di#usion equation by replacing the second-order space derivative with a fractional Riesz derivative of order  ∈ (0; 2] and the Grst-order time derivative with a fractional Caputo derivative of order  ∈ (0; 1]. Choosing the initial condition in analogy to that fulGlled for (1.1), we write

For use in Section 2 it is important to observe that the master equation (1.1) holds also for generalized functions p, and w representing probability measures. We do not hesitate to label such functions as “probability densities”.

0 ¡  6 1; s ¿ 0;

(1.7)

2 Let us recall that a generic linear pseudo-di#erential operator A, acting with respect to the variable ∈ R, is deGned through its  +∞ xix ˆ ˆ f(), e A[f(x)] d x = A() Fourier representation, namely −∞ ˆ where A() is referred to as symbol of A, formally given as ˆ A() = (Ae−ix )e+ix . In previous papers we have denoted the operator x R by D0 or by x D0 .

R. Goren2o, A. Vivoli / Signal Processing 83 (2003) 2411 – 2420

hence turns out to be deGned as, see e.g. [10],

and we obtain immediately s−1 ; s ¿ 0;  ∈ R: u(; ˜ˆ s) =  s + ||

 t D∗ f(t)

:=

    

1 (1 − )

  d   f(t); dt


0

t

f(1) () d; (t − )

2413

0 ¡  ¡ 1;  = 1: (1.8)

It can alternatively be written in the form
t d f() 1  d t D∗ f(t) = (1 − ) dt 0 (t − ) t − f(0+ ) (1 − )
t d f() − f(0+ ) 1 d; = (1 − ) dt 0 (t − )

(1.11)

By our derivation of (1.4) from (1.1) in [14] we have de-mystiGed the often asked-for meaning of the time fractional derivative in the fractional di#usion equation. We have shown there that the fractional derivatives are caused by asymptotic power laws and well-scaled passage to the di#usion limit. Let us cite in this context [2]. This author discusses appealing variants, namely evolution, discrete only in time or only in space, of densities under special power-law regimes. By his way he arrives in the di#usion limit at the particular cases (a) 0 ¡  6 2,  = 1, (b)  = 2, 0 ¡  6 1.

+

0 ¡  ¡ 1:

2. Aim of this essay

(1.9)

The Caputo derivative has been indexed with ∗ in order to distinguish it from the classical Riemann– Liouville fractional derivative t D , the Grst term at the R.H.S. of the Grst equality in (1.9). As it can be noted from the last equality in (1.9) the Caputo derivative provides a sort of regularization at t = 0 of the Riemann–Liouville derivative; for more details see [10]. The space–time fractional di9usion equation (1.4) contains as particular cases the space fractional di9usion equation when 0 ¡  ¡ 2 and =1, the time fractional di9usion equation when  = 2 and 0 ¡  ¡ 1, and the standard di9usion equation when  = 2 and  = 1. We note that for the fractional cases the word “di#usion” is also justiGed because the fundamental solution (or the Green function) in all cases can be interpreted as a space probability density evolving in time, see e.g. [9,20,21]. In the Fourier–Laplace domain the Cauchy problem for the space–time fractional di#usion equation (1.4) appears in the form s u(; ˜ˆ s) − s−1 = −|| u(; ˜ˆ s); 0 ¡  6 2; 0 ¡  6 1

In [14] we have assumed that the jump width pdf w(x) is an even function (w(x) = w(−x)) and has a Gnite second moment (variance) or exhibits the asymptotic behaviour w(|x|) ∼ b|x|−(+1) with some , 0 ¡  ¡ 2, for |x| → ∞, and the waiting time pdf (t) has a Gnite Grst moment (mean) or exhibits the asymptotic behaviour (t) ∼ ct −(+1) with some , 0 ¡  ¡ 1, for t → ∞. The coeKcients b and c in these asymptotics are positive constants. In discrete analogy we now, contrastingly, set ∞  pk (x − k); w(x) = k=−∞

(t) =

cn (t − n);

(2.1)

n=1

whence for the transforms of Fourier and Laplace we obtain ∞ ∞   ik ˜ w() ˆ = pk e ; (s) = cn e−ns : (2.2) n=1

k=−∞

Naturally, For all pk ¿ 0;

∞ 

pk = 1

for all cn ¿ 0;

k=−∞ ∞ 

(1.10)

∞ 

n=1

cn = 1:

(2.3)

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R. Goren2o, A. Vivoli / Signal Processing 83 (2003) 2411 – 2420

We further assume symmetry: pk = p−k for all k. We are interested in a random walk that proceeds as follows. A particle just having jumped in the time instant t (=a non-negative integer) to the space point x (=an integer) remains sitting there until time instant t + n− , that is just before the instant t + n. Here n, the waiting time, is a natural number ¿ 1. The probability for this waiting time is cn . Then in the instant t + n it jumps with probability pk from the point x to the point x + k where k is an integer. To say it more precisely: in the initial instant t = 0 the particle is sitting in the origin x = 0. Whereas in [14] the waiting times T = Tj are positive real numbers, here they are positive integers nj , j =1; 2; 3; : : : : The jumps occur in the instants t1 = n1 , t2 = n1 + n2 , t3 = n1 + n2 + n3 ; : : : : Again we assume that the waiting times are iid and that the jumps widths are iid, and that waiting times and jumps are independent of each other. It will turn out that under appropriate assumptions on asymptotics by well-scaled compression of waiting times and jump widths in the di#usion limit the resulting process obeys the fractional di#usion equation (1.4) for the t-dependent sojourn probability density u(x; t) of the particle being in point x. For this we need to state two lemmata, that extend to the fully discrete case the corresponding ones proved for the continuous case in [14]. Note that by (2.1) we have no longer a CTRW, but a random walk discrete both in space and in time, which we refer to as fully discrete random walk (FDRW). +∞ Lemma 1. Assume all pk ¿ 0, k=−∞ pk = 1, symmetry pk = p−k for all integers k and either (a) or (b): +∞ 

(a) #2 :=

k 2 pk ¡ ∞

labelled as  = 2;

k=−∞

(b) pk ∼ bk −(+1)

for k → ∞

with  ∈ (0; 2)

and b ∈ R+ : Then, with +=

#2 2

+=

b, ( + 1) sin(,=2)

we have the asymptotics w() ˆ = 1 − +|| + o(|| ) for  → 0. Lemma 2. Assume all cn ¿ 0, (A) or (B): (A) - :=

in case (b);

ncn ¡ ∞

n=1 cn =1, and either

labelled as  = 1;

n=1

(B) cn ∼ cn−(+1) ;

for n → ∞

with  ∈ (0; 1)

and c ∈ R+ : Then, with . = - in case (A), . = c(1 − )= in case (B), we have the asymptotics ˜ (s) = 1 − .s + o(s ) for s → 0+ . Remark. In [14] we have h and s in the asymptotic formulas in place of  and s. But this is a trivial replacement, needed for application of the continuity theorems for sequences of characteristic functions and sequences of Laplace transforms. By o(z) for z → 0 we mean o(z)=z → 0 as z → 0 (Landau notation). The proofs of these lemmata can be extracted from considerations in the papers [6,13]. Essentially, these lemmata express the fact that the sequence of the probabilities pk is lying in the domain of attraction of the symmetric stable law with index , whereas the sequence of probabilities cn is lying in the domain of the extreme (right-sided) stable law with index . See also Gnedenko’s important theorem in [7] (which in its formulation unfortunately has a misprint in a constant). Now, by replacing the waiting times T = n and the jumps X = k by T = n and hX = hk with the positive scaling factors  and h we obtain a random walk on the spatial grid {kh | k = : : : − 3; −2; −1; 0; 1; 2; : : :} starting in the origin x = 0 at instant t = 0 and proceeding through the instances n, n=1; 2; 3; : : : : Then, understanding the master equation (1.1) in the sense of generalized functions, working with the combined transforms of Fourier and Laplace and using our two lemmata we arrive, by introducing the scaling relation . = +h

in case (a);

∞ 

∞

(2.4)

and letting h and  tend to zero, in the limit at the Cauchy problem (1.4) for the space–time fractional di#usion equation. The way of reasoning is the same

R. Goren2o, A. Vivoli / Signal Processing 83 (2003) 2411 – 2420

as in [14,15]. In di#erent notation, essentially the same way of scaling has been applied in the recent paper [26].

3. Examples We present some examples by giving the jump width probabilities pk and the waiting time probabilities cn , then identifying the adjustment constants b and c, and the scaling constants . and +. First example: the pure power random walk. Take, with 0 ¡  ¡ 2, 0 ¡  ¡ 1, Z=the set of all integers, N = the set of positive integers, p0 = 0; pk = b|k|−(+1) cn = cn−(+1)

(3.1)

We identify readily (with 0(z) denoting the Riemann zeta function) 1 ; 0( + 1)

b=

1 ; 20( + 1)

+=

, ; 20( + 1)( + 1) sin(,=2)

.=

(1 − ) : 0( + 1)

c=

  

n

 



=

; n ∈ N; 0 ¡  6 1:

n ∞ Then n=1 cn = 1 and, by Lemma 2,

cn ∼ cn−(+1)

with c = ( + 1)

(3.4)

|sin(,)| : ,

(3.5)

Furthermore (compare [17]) c1 = 1 and cn = 0

for n ¿ 2 in the case  = 1; if 0 ¡  ¡ 1

and, with respect to Lemma 2, . = 1 for both cases (A) and (B). Remark. For 0 ¡  ¡ 1 the cn form a completely monotone sequence. In fact, setting T0 cn = cn ; Tcn = cn −cn+1 , generally Tj+1 cn =Tj cn −Tj cn+1 for j ¿ 0 we Gnd byinduction, adopting the empty product conn vention ( j=m aj = 1 if n ¡ m), k

n−1

j=0

j=1

  1 T cn = (j + ) (j − ) ¿ 0 (n + k)! k

(3.2)

Remark. The sequences p1 ; p2 ; p3 ; : : : and c1 ; c2 ; c3 ; : : : have the nice property of being completely monotone. The boundary cases  = 2 and  = 1 are singular because of divergence of the series in (a) and (A). Second example: the binomial (or GrSunwald– Letnikov) random walks. From a footnote in [11] we take: Lemma 3. For positive non-integer 1 we have the asymptotic relation

 

1 |sin(1,)| −(1+1)

k

∼ (1 + 1)

k , for k → ∞:

cn = (−1)

n+1

c1 ¿ c2 ¿ c3 ¿ · · · → 0

for 0 = k ∈ Z;

for n ∈ N:

Take

2415

(3.3)

for all n ∈ N; k ∈ {0} ∪ N: With respect to the jump probabilities we distinguish, compare with [8,11,13], the cases (aa) 0 ¡  ¡ 1;

(bb) 1 ¡  ¡ 2;

(cc)  = 2

leaving the singular case  = 1 to a separate treatment. We take in case (aa): 2 p0 = 1 − ; cos(,=2)    2 |k|+1 for k = 0; pk = (−1) 2 cos(,=2) |k| in case (bb): p0 = 1 +

2 ; cos(,=2)

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R. Goren2o, A. Vivoli / Signal Processing 83 (2003) 2411 – 2420

2 p1 = p−1 = − 2 cos(,=2) pk = p−k = (−1)k  ×





 1+

   2

Remark. In the second and third examples the waiting time probabilities cn vanish for n ¿ 2 in the case  = 1. In this case the resulting random walks have no memory, they represent Markov chains.

;

2 2 cos(,=2)

4. Numerical results and conclusions

for k ¿ 2:

k +1

For the parameter 2 we require the restriction 0 ¡ 2 6 cos(,=2) 0¡26

in case (aa);

1 |cos(,=2)| 

in case (bb):

Note that in contrast to [13] we here have replaced the letter + by the letter 2, having already used up the letter + in Lemma 1. Now, using Lemma 3, we identify in case (aa), as well as in case (bb), 2 b= 2(1 − ) cos(,=2) =

2 ( + 1) sin(,=2); ,

+ = 2;

(3.6)

so that our original notation of [13] is vindicated. In the remaining case (cc), namely  = 2 the above formulas given for (bb) still hold, but reduce to p0 = 1 − 22;

p1 = p−1 = 2;

pk = 0

for |k| ¿ 2

(3.7)

with the restriction 0 ¡ 2 6 12 , and so, with (a) in Lemma 1, we Gnd #2 = 22 and again + = 2. Third example: In the formulas above for cases (aa) and (bb) the value  = 1 leads to vanishing denominator, so we need another set of jump probabilities pk whereas we keep the waiting time probabilities cn of the second example, see (3.4). As in [12] we take (again + replaced by 2) p0 = 1 −

22 ; ,

pk =

, with 0 ¡ 2 6 : 2

2 ,|k|(|k| + 1)

for k = 0 (3.8)

Then, pk ∼ (2=,)k −2 and we identify b = 2=, and Gnd + = b,=(2) sin(,=2) = 2.

In this section, we show the graphical results of Monte-Carlo simulations of our FDRW described in Examples 1–3. We have considered six di#erent case studies for which the relevant parameters are reported in Table 1. According to Section 2 by a proper scaling of the jump widths and waiting times, in our simulations we expect that the probability density functions of the stopping points, namely the points hSN , where N is the integer deGned by tN ¡ 1 6 tN +1 , tend to the corresponding solutions of the fractional di9usion equation (1.4) in the limit for h → 0 keeping true the scaling relation (2.4). In practice we have rescaled the length and time units of the processes using a Gnite (but small) value of h (given in Table 1) and the corresponding value of  according to (2.4). The results of the simulations of the sample paths and histograms for our 6 case studies are reported in Figs. 1–6. On the left-hand side of the Ggures we have presented sample paths, resulting from an FDRW of duration of 104 time units. The number J of jumps occurred for each case is shown in Table 1. On the right-hand side of the Ggures we have shown the histograms realized with 10,000 rescaled simulations according to the previous considerations, comparing them with the fundamental solutions (plotted in the Ggures) of the fractional di#usion equation. These solutions have been obtained by numerically inverting their Fourier transforms (the characteristic functions) by using the relevant formulas and the asymptotic expressions exhibited in [20]. To obtain the marker points of the histograms, we have divided the real axis in 100 intervals, chosen in such a way that the corresponding probabilities equal 1 100 . So we expect that in each of these intervals fall about 100 (rescaled) stopping points. We have then counted how many stopping points of our simulations fall in each interval (bin), and we have divided the relative frequency by the length of the interval (in order to get normalized histograms).

R. Goren2o, A. Vivoli / Signal Processing 83 (2003) 2411 – 2420

2417

Table 1 The relevant parameters for the simulations Example



1 2 2 2 2 3



1.5 0.97 1.5 2 2 1

0.85 0.95 0.95 0.95 1 1

J

h

2

+

.

565 7107 7505 7368 9999 9999

10−2

— 3 × 10−3 2:99 × 10−1 4:20 × 10−2 3:83 × 10−1 8:69 × 10−1

1.25 3 × 10−3 2:99 × 10−1 4:20 × 10−2 3:83 × 10−1 8:69 × 10−1

4.04 1 1 1 1 1

10−2 10−2 2 × 10−2 2:82 × 10−2 10−2

0

10

100

EX 1 α = 1.5 β = 0.85

50 -1

10 pdf

S

N

0 -50

10

α = 1.5 β = 0.85

-150 -200

-2

EX 1

-100

-3

0

2000

4000

6000

8000

10

10000

6

4

2

0

2

4

6

hS

t

N

Fig. 1. A sample path (left) and the histogram (right) with plotted fundamental solution in Example 1 with { = 1:5;  = 0:85}.

10

0

5

EX 2

0

EX 2

5

α = 0.97 β = 0.95

10

-1

α = 0.97 β = 0.95

pdf

SN

-10 -15 -20

10

-2

-25 -30 -35

0

2000

4000

6000

8000

10000

t

10

-3

6

4

2

0

2

4

6

hS N

Fig. 2. A sample path (left) and the histogram (right) with plotted fundamental solution in Example 2 with { = 0:97;  = 0:95}.

We have drawn the marker points as dots in the Ggures in correspondence to the mean value of the (scaled) stopping points hSN fallen in each bin, limiting ourselves to the interval |hSN | 6 6.

We point out that all programs have been written in MATLAB 5. In our earlier papers [15–17] we have discretized the time fractional derivative in the di#usion equation

2418

R. Goren2o, A. Vivoli / Signal Processing 83 (2003) 2411 – 2420 10

250

EX 2

EX 2

200

150 α = 1.5 β = 0.95 100

10

50

α = 1.5 β = 0.95

-1

pdf

SN

0

0

10

-50

-2

-100 -150 -200

10 0

2000

4000

6000

8000

-3

10000

6

4

2

0

2

4

6

hS

N

t

Fig. 3. A sample path (left) and the histogram (right) with plotted fundamental solution in Example 2 with { = 1:5;  = 0:95}.

0

10 30

EX 2

20

EX 2

10

α=2 β = 0.95

α=2 β = 0.95

-1

10

S

N

pdf

0 10

-2

10 20 30 40

-3

0

2000

4000

6000

8000

10

10000

6

4

2

t

0 hSN

2

4

6

Fig. 4. A sample path (left) and the histogram (right) with plotted fundamental solution in Example 2 with { = 2;  = 0:95}.

0

10

40

EX 2 20 -1

α=2 β=1

10

S

N

pdf

0

-20

-2

10 EX 2 -40

-60

α=2 β=1 -3

0

2000

4000

6000 t

8000

10000

10

-6

-4

-2

0

2

4

6

hS

N

Fig. 5. A sample path (left) and the histogram (right) with plotted fundamental solution in Example 2 with { = 2;  = 1}.

R. Goren2o, A. Vivoli / Signal Processing 83 (2003) 2411 – 2420 10

2419

0

3000

EX 3 EX 3

2500 α=1 2000 β = 1

-1

pdf

10

α=1 β=1

S

N

1500 1000

10

-2

500 0 -500

0

2000

4000

6000

8000

10000

10

-3

-6

-4

-2

0

2

4

6

hS

t

N

Fig. 6. A sample path (left) and the histogram (right) with plotted fundamental solution in Example 3 with { = 1;  = 1}.

(1.4) via the GrSunwald–Letnikov scheme backwards until the initial instant t = 0. We have obtained fully discrete random walks of the memory type, remarkably with the same binomial probabilities cn as here in our second example. In the present paper, however, we have forwardoriented random walks in which, after every jump (as in the classical continuous-time random walk) the whole past is forgotten. It is desirable that comparative numerical studies are carried out in order to Gnd out the relative merits of di#erent choices of the probabilities for waiting times and jumps. Some numerical experiments (not reported here) and our intuition let us expect that near the critical values 0, 1 and 2 for , 0 and 1 for , the quality of approximation of the scheme in question needs careful consideration. Convergence (in distribution) we always have, but its speed deserves further research.

Acknowledgements This work has been carried out in the framework of the Erasmus–Socrates project (between the Universities of Berlin and Bologna) and of the INTAS project 00-0847. The authors are very grateful to Prof. Francesco Mainardi for the inspiring discussions and the helpful comments: they dedicate this contribution to him on the occasion of his 60th birthday. We appreciate the helpful suggestions of the anonymous referees.

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