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4.2.3
Fractional operator walk
The simple random walk has provided a way to understand the Gaussian distribution
with its aggregation of a large number of identically distributed random variables having
finite variance. This is the background for the law of frequency of errors discussed in the
first chapter. More general statistical phenomena can be modeled by means of fractals,
and we considered the case in which the jump-size distribution function is an inverse
power law. The spatial heterogeneity modeled by inverse power-law jump probabilities
gives rise to the class of Lévy distributions of which the Gaussian is the unique member
having a finite variance. Another way to model the complexity of dynamic webs is
through non-integer differential and integral operators, which we now develop in the
random-walk context.
The idea of a fractional derivative or integral was strange when it was first introduced;
perhaps it is still strange, but at least it is well defined and even useful. In an analogous
way the notion of a fractional difference can be developed to incorporate long-term
memory into the random walks we have discussed. To develop fractional differences
we introduce the shift operator
B
defined by the operation
BQ
j
+
1
=
Q
j
(4.42)
which shifts the data from element
1toelement
j
; that is,
B
shifts the process
Q
j
from its present value to that of one time step earlier. Using this operator a simple
random walk can be formally written as
j
+
(
−
)
Q
j
+
1
=
ξ
j
,
1
B
(4.43)
where
ξ
j
is the discrete random force. Note that this is a discrete analog of the Langevin
equation for a free particle with the time step set to unity, with a solution given by
N
Q
N
=
1
ξ
j
,
(4.44)
j
=
where the equal-magnitude contributions for the Bernoulli walk are replaced by inde-
pendent identically distributed random variables. If the second moment of the random
fluctuations is finite the CLT assures us that the statistical distribution of the sum will
be Gaussian.
Hosking [
24
] generalized (
4.43
) to the fractional difference equation
)
α
Q
j
+
1
=
ξ
j
,
(
−
(4.45)
1
B
where the power-law index
need not be an integer. We assert here without proof that
equations of the fractional random-walk form are the direct analog of the fractional
Langevin equation discussed in Chapter 5 in the same way that the traditional random
walks are direct analogs of the ordinary Langevin equations discussed in Chapter 3. The
question remains that of how to operationally interpret the formal expression (
4.45
).
α
