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−
ψ(
1
u
)
P
∗
(
k
,
u
)
=
u
1
)
.
(5.133)
)ψ(
−
p
(
k
u
This is the Montroll-Weiss equation for the standard CTRW.
The further interpretation of (
5.133
) is accomplished by restricting the solution to the
situation that the walker starts from the origin so that
P
, and conse-
quently the probability density function is the propagator for the CTRW process. This
interpretation is more apparent in the space-time form of the phase-space equation, so
we introduce the Laplace transform of the memory kernel,
(
q
,
t
=
0
)
=
δ(
q
)
u
ψ(
)
u
φ(
)
=
)
,
u
(5.134)
−
ψ(
1
u
into (
5.133
), which, after some algebra, yields
1
P
∗
(
k
,
u
)
=
)
]
.
(5.135)
+
φ(
u
u
)
[
1
−
p
(
k
Equation (
5.135
) can be put into the simpler form
uP
∗
(
=−
φ(
P
∗
(
k
,
u
)
−
1
u
)
[
1
−
p
(
k
)
]
k
,
u
),
(5.136)
whose inverse Fourier-Laplace transform yields the integro-differential equation for the
propagator
t
p
d
q
∂
P
(
q
,
t
)
dt
φ(
t
)
t
)
+
q
)
q
,
t
)
=
t
−
−
P
(
q
,
(
q
−
P
(
.
(5.137)
∂
t
0
Equation (
5.137
) is the Montroll-Kenkre-Shlesinger master equation [
10
] and it is
clearly non-local in both time and space. In time the non-locality is determined by the
memory kernel, which in turn is determined by the waiting-time distribution function,
that is, the inverse Laplace transform of (
5.134
). The spatial non-locality is determined
by the jump probability, which is the inverse Fourier transform of the structure function.
Equation (
5.133
) is the simplest situation of a factorable random walk, when the
origin of time coincides with the beginning of the waiting time. The Fourier-Laplace-
transformed probability can be used as the moment-generating function. Recall that the
Fourier transform of the probability density is the characteristic function. For example,
the first moment of the process, in one dimension, can be written as
k
=
0
.
P
∗
(
i
∂
k
,
u
)
LT
{
q
;
t
;
u
}= −
(5.138)
∂
k
Inserting (
5.133
)into(
5.138
) and taking the appropriate derivatives yields the Laplace
transform of the first moment,
μ
1
ψ(
u
)
LT
{
q
;
t
;
u
}=
u
1
)
.
(5.139)
−
ψ(
u
The first moment is obtained from the definition of the structure function
k
=
0
=
i
∂
(
)
p
k
μ
1
=−
(
).
qp
q
(5.140)
∂
k
q
