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The formal solution to ( 3.131 ) can be obtained by factoring the solution as
P
(
q
,
t
|
q 0 ) =
W
(
t
)
F
(
q
,
q 0 )
to obtain the eigenvalue equation
1
W
dW
dt =
1
F L FP F
=− γ.
(3.133)
Upon indexing the eigenvalues in order of increasing magnitude
| γ 0 | < | γ 1 | <...
we
obtain the solution as the expansion in eigenfunctions
φ j (
q
,
q 0 )
of the Fokker-Planck
operator
e γ j t
P
(
q
,
t
|
q 0 ) =
φ j (
q
,
q 0 ).
(3.134)
j
The FPE has been the work horse of statistical physics for nearly a century and some
excellent texts have been written describing the phenomena to which it can be applied.
All the simple random walks subsequently discussed can equally well be expressed in
terms of FPEs. However, when the web is complex, or the environment has memory, or
when any of a number of other complications occur, the FPE must be generalized. Even
in the simple case of a dichotomous process the statistics of the web's response can be
non-trivial.
Multiplicative fluctuations
The above form of the FPE assumes that the underlying dynamical web is thermo-
dynamically closed; that is, the fluctuations and dissipation have the same source and
are therefore related by the FDR. An open stochastic web, in which the sources of the
fluctuations and any dissipation in the web may be different, might have a dynamical
equation of the form
dQ
(
t
)
=
G
(
Q
) +
g
(
Q
)ξ(
t
),
(3.135)
dt
where G
are analytic functions of their arguments. If we again assume
that the fluctuations have Gaussian statistics and are delta-correlated in time then the
Stratonovitch version of the FPE is given by
( · )
and g
( · )
P
P
(
q
,
t
|
q 0 )
=
)
G
(
q
) +
Dg
(
q
q g
(
q
)
(
q
,
t
|
q 0 ).
(3.136)
t
q
There is a second probability calculus, due to Itô, which is completely equivalent to that
of Stratonovitch, but we do not discuss that further here. Solving ( 3.136 ) in general is
notoriously difficult, but it does have the appealing feature of yielding an exact steady-
state solution.
The steady-state solution to ( 3.136 ) is obtained from the condition
P ss (
P ss (
q
)
=
)
=
0
G
(
q
) +
Dg
(
q
q g
(
q
)
q
),
(3.137)
t
q
 
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