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An alternative procedure can be developed for determining the first-passage-time pdf
through the application of the theory of a stochastic process on a line bounded on one
side by an absorbing barrier. Assuming that the absorbing barrier is located at the point
q
a
, the probability that a particle survives until time
t
without being absorbed is
q
a
F
(
q
a
,
t
)
=
P(
q
,
t
)
dq
.
(3.178)
−∞
The quantity
dq
)
at time
t
in the presence of the absorbing barrier. The probability of its being absorbed
in the time interval
P(
q
,
t
)
dq
is the probability of the particle being in the interval (
q
,
q
+
(
t
,
t
+
δ
t
)
is
dF
(
)
dt
δ
q
a
,
t
F
(
q
a
,
t
)
−
F
(
q
a
,
t
+
δ
t
)
=−
t
(3.179)
and consequently the first-passage-time pdf can be expressed as
q
a
dF
(
q
a
,
t
)
d
dt
W
(
q
a
,
t
)
=−
=−
P(
q
,
t
)
dq
.
(3.180)
dt
−∞
It is a straightforward matter to rederive (
3.176
) starting from (
3.180
) using the unbiased
Gaussian distribution.
The first-passage-time pdf for a homogeneous diffusion process can be determined
using the Kolmogorov backward equation [
16
]
2
P
∂
P
(
q
,
t
;
q
0
)
q
0
)
∂
P
(
q
,
t
;
q
0
)
q
0
)
∂
(
q
,
t
;
q
0
)
=
a
(
+
b
(
,
(3.181)
q
0
∂
t
∂
q
0
∂
where the derivatives are being taken with respect to the initial condition
Q
(
0
)
=
q
0
.
The Laplace transform of this equation with
q
a
>
q
>
q
0
gives
q
0
)
∂
P
P
2
(
q
,
u
;
q
0
)
q
0
)
∂
(
q
,
u
;
q
0
)
u P
(
q
,
u
;
q
0
)
=
a
(
+
b
(
,
(3.182)
q
0
∂
q
0
∂
so that using the ratio of probabilities in (
3.172
) yields the following second-order
differential equation for the Laplace transform of the first-passage-time distribution:
W
q
0
)
∂
W
2
q
0
)
∂
(
q
,
u
;
q
0
)
(
q
,
u
;
q
0
)
u W
b
(
+
a
(
−
(
q
,
u
;
q
0
)
=
0
.
(3.183)
q
0
∂
q
0
∂
Feller [
16
] proved that the condition (
3.176
) and the requirement that
W
(
q
a
,
u
;
q
0
)<
∞∀
q
0
(3.184)
together determine that the solution to (
3.183
) is unique.
3.4
More on statistics
The increase in complexity of a web can be achieved by extending the Hamiltonian
to include nonlinear interactions, as we do later. Another way to increase complexity
is to put the web in contact with the environment, which in the simplest physical case
