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density
p
has a finite second moment and is symmetric; the randomwalk is otherwise
rather insensitive to its precise form.
We now investigate some consequences of having a single-step probability density
that has diverging central moments and find that it leads us to the fractional calculus.
(
q
)
4.2.1
The Weierstrass walk
The assumption that the mean-square step length diverges as
q
2
q
2
p
=
(
q
)
=∞
(4.20)
q
is not sufficient to specify the form of the random walk, since the transition probability
leading to this divergence is not unique. Consider a jump distribution that leads to non-
diffusive behavior,
∞
a
n
δ
q
,
b
n
+
δ
q
,
−
b
n
,
−
a
1
1
p
(
q
)
=
(4.21)
2
a
n
=
0
which was introduced by Hughes
et al
.[
25
], where
b
and
a
are dimensionless constants
greater than one. The mean-square jump length is then given by
b
2
a
n
∞
a
−
1
q
2
=
,
(4.22)
a
n
=
0
which for
b
2
a
is infinite since the series diverges. The lattice structure function is the
discrete Fourier transform of the jump probability and is the analog of a characteristic
function
>
∞
∞
a
−
1
1
a
n
cos
e
iks
p
s
=
b
n
k
p
k
=
(
)
(4.23)
a
s
=−∞
n
=
0
and for obvious reasons was called the Weierstrass walk by Hughes
et al
.[
25
]. The
divergence of the second moment is a consequence of the non-analytic properties of the
Weierstrass function.
Consider the physical interpretation of the process defined by the jump distribution
(
4.21
) wherein the probability of taking a step of unit size is 1
/
a
, the probability of
taking a step a factor
b
larger is a factor 1
a
smaller and so on, with increasingly larger
steps occurring with increasingly smaller probabilities. This is a Bernoulli scaling pro-
cess that emerged in the resolution of the St. Petersburg paradox, where the player tries
to determine a characteristic scale from a process that does not possess such a scale. If
a billion people play the game, then, on average, half will win one coin, a quarter will
win two coins, an eighth will win four coins, and so on. Winnings occur on all scales,
just as do the wiggles in the Weierstrass function, which is the paradigm of fractals.
So how does this winning an order of magnitude more money (in base
b
) occurring
with an order-of-magnitude-lower probability (in base
a
) translate into a random walk?
As the Weierstrass walk progresses the set of sites visited by the walker consists of
localized clusters of sites, interspersed by gaps, followed by clusters of clusters over a
/
