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probability satisfies the normalization condition
p
(
q
,
t
)
=
ψ(
t
),
(5.126)
q
where here, as before,
is the waiting-time distribution density and the integral over
time yields the overall normalization
∞
ψ(
t
)
∞
0
ψ(
p
(
q
,
t
)
dt
=
t
)
dt
=
1
.
(5.127)
0
q
Montroll and Weiss made the specific assumption that the length of a pause and the
size of a step are mutually independent, an assumption we use temporarily, but avoid
making later in the discussion. For the moment, assuming space-time independence in
the jump-length probability yields the product of the waiting-time distribution density
and the jump-length probability
(
,
)
=
(
)ψ(
),
p
q
t
p
q
t
(5.128)
in which case the random walk is said to be factorable. The Fourier-Laplace transform
of the transition probability yields
e
i
k
·
q
∞
0
p
∗
(
e
−
ut
p
)ψ(
k
,
u
)
≡
(
q
,
t
)
dt
=
p
(
k
u
),
(5.129)
q
where
denotes both the discrete and the continuous Fourier transforms of the
jump probability. We use
W
∗
to denote the double Fourier-Laplace transform of
W
throughout. Consequently, using the convolution form of (
5.125
), we obtain
p
(
k
)
p
0
(
k
,
u
)
W
∗
(
k
,
u
)
=
(5.130)
1
−
p
∗
(
k
,
u
)
and for the double transform of the probability from (
5.124
) we have, using (
5.130
),
ˆ
(
p
0
(
u
)
k
,
u
)
P
∗
(
)
=
ˆ
0
(
k
,
u
u
)
+
)
.
(5.131)
1
−
p
∗
(
k
,
u
In the case in which the transition from the origin does not play a special role
p
0
(
,
0
(
)
=
(
p
∗
(
k
,
u
)
=
k
,
u
)
u
u
)
and (
5.131
) simplifies to
(
u
)
P
∗
(
k
,
u
)
=
)
,
(5.132)
p
∗
(
1
−
k
,
u
which can be further simplified when the space-time transition probability factors. The
relation between the survival probability
and the waiting-time distribution density
isgivenby(
5.51
) and the relation between the Laplace transform of the two quantities
by (
5.57
). Consequently, the factorable network (
5.132
) simplifies to
(
t
)
