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from
q
to
q
2
in the time remaining (
t
), not necessarily for the first time. Then we
can write the integral equation for the pdfs
−
τ
t
P
(
q
2
−
q
1
,
t
)
=
P
(
q
2
−
q
,
t
−
τ)
W
(
q
−
q
1
,τ)
d
τ
(3.170)
0
and the function
W
τ.
The convolution form of (
3.170
) allows use of Laplace transforms to obtain the
product form
(
q
,τ)
is the first-passage-time pdf for the transition 0
→
q
in time
P
)
=
P
)
W
(
q
2
−
q
1
,
u
(
q
2
−
q
,
u
(
q
−
q
1
,
u
),
(3.171)
so that the Laplace transform of the first-passage-time pdf is
P
(
q
2
−
q
1
,
u
)
W
(
q
−
q
1
,
u
)
=
)
.
(3.172)
P
(
q
2
−
q
,
u
What is now necessary is to take the inverse Laplace transform of the ratio of the two
point pdfs.
As an example of the application of this general formula let
P
be the solution
to the FPE for a diffusive process, that is, the unbiased Gaussian transition probability
(
q
,
t
)
exp
∞
q
2
4
Dt
1
1
2
e
−
ikq
e
−
Dtk
2
dk
P
(
q
,
t
)
=
√
4
−
=
.
(3.173)
π
π
Dt
−∞
The next step is to take the Laplace transform of this pdf to obtain
∞
1
2
P
e
−
ikq
Dk
2
)
−
1
dk
(
q
,
u
)
=
(
u
+
π
−∞
exp
2
1
/
1
2
√
Du
u
D
=
−|
q
|
.
(3.174)
Finally, by taking into account the ordering of the variables in (
3.172
) we can write
exp
2
1
/
u
D
W
(
q
−
q
1
,
u
)
=
−
(
q
−
q
1
)
,
(3.175)
so that the inverse Laplace transform of (
3.175
) yields
t
3
/
2
exp
2
q
−
q
1
1
−
(
q
−
q
1
)
W
(
q
−
q
1
,
t
)
=
√
4
,
for
t
>
0.
(3.176)
4
Dt
π
D
Note that this pdf does not have a finite first moment, as can be shown using
∞
∞
0
∂
e
−
ut
W
=
t
tW
(
q
,
t
)
dt
=−
lim
u
(
q
,
t
)
dt
∂
u
→
0
0
0
∂
W
(
q
−
q
1
,
u
)
=−
lim
u
,
(3.177)
∂
u
→
so th
a
t, upon inserting (
3.175
)into(
3.177
), the first-passage-time moment diverges as
1
/
√
u
as
u
→
0
.
