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and consequently the convolution form of (
7.40
) reduces to the product of Laplace
transforms
ψ
n
(
)
=
ψ
n
−
1
(
)ψ
1
(
u
u
u
),
so that substituting from (
7.42
) and carrying out the iteration yields (
3.193
)
=
ψ(
)
n
ψ
n
(
u
)
u
.
(7.43)
The Laplace transform of the survival probability
(
t
)
is
−
ψ(
1
u
)
(
u
)
=
,
(7.44)
u
so that inserting (
7.43
) and (
7.44
) into the Laplace transform of (
7.39
) and again using
the convolution theorem yields
∞
−
ψ(
ψ(
)
n
p
1
u
)
p
(
u
)
=
u
)(
I
+
K
(
0
).
(7.45)
u
n
=
0
1 and
ψ(
1, having the value
ψ(
The eigenvalues of the matrix
(
I
+
K
)
are
≤
u
)<
u
)
=
1
only in the limiting case
u
0. Thus, we evaluate the geometric sum appearing in (
7.45
)
according to the well-known rule
=
∞
ψ(
)
n
1
u
)(
I
+
K
=
)
.
(7.46)
−
ψ(
1
u
)(
I
+
K
n
=
0
Consequently, inserting this form of the series into (
7.45
) gives us for the Laplace
transform of the probability density
−
ψ(
−
ψ(
1
u
)
1
u
)
p
(
u
)
=
p
(
0
)
=
p
(
0
)
u
ψ(
−
ψ(
u
ψ(
u
ψ(
u
−
u
)(
I
+
K
)
u
(
1
u
))
+
u
)
−
u
)(
I
+
K
)
and, using the form of the unit matrix, this reduces to
−
ψ(
1
u
)
p
(
u
)
=
K
p
(
0
).
(7.47)
−
ψ(
u
ψ(
u
(
1
u
))
−
u
)
−
ψ(
By dividing both the numerator and the denominator of (
7.47
)by
(
1
u
))
we obtain
1
p
(
u
)
=
K
p
(
0
),
(7.48)
−
(
u
u
)
where
u
ψ(
u
)
(
u
)
=
)
,
(7.49)
−
ψ(
1
u
namely the typical memory kernel that emerged in CTRW theory discussed previously.
With a little algebra (
7.48
) can be rewritten as
)
=
(
u
p
(
u
)
−
p
(
0
u
)
K
p
(
u
),
(7.50)
