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but
1 by definition for the survival probability. The waiting-time distribu-
tion density is the negative time derivative of the survival probability so that (
7.75
)
simplifies to
(
0
)
=
t
d
(
t
)
dt
R
t
)ψ(
t
)ξ
p
(
t
).
=
ε
R
(
t
)ξ
p
(
t
)
−
ε
(
t
−
(7.76)
dt
0
Equation (
7.76
) ought to be exactly the same as the perturbed GME (
7.64
) if, in fact,
(
7.74
) is a solution to the GME. Assume (
7.74
) to be true and equate the integral terms
of the two equations to obtain
t
t
dt
R
t
)ψ(
t
)ξ
p
(
t
)
=−
dt
(
t
)(
t
),
ε
(
t
−
t
−
(7.77)
0
0
the validity of which must be established. Inserting the assumed form for the solution
(
t
)
into (
7.77
) yields the rather awkward-looking expression
t
t
t
t
).
(7.78)
The Laplace transform of (
7.78
) together with judicious use of the convolution theorem
leads to the expression
dt
R
t
)ψ(
t
)ξ
p
(
t
)
=
ε
dt
(
t
)
dt
R
t
)(
t
−
t
)ξ
p
(
ε
(
t
−
t
−
(
0
0
0
−
ψ(
)LT
R
u
=
ε(
LT
R
u
1
u
)
εψ(
u
(
t
)ξ
p
(
t
)
;
u
)
(
t
)ξ
p
(
t
)
;
(7.79)
u
and cancelling out common terms on the right- and left-hand sides of this equation
leads to
u
ψ(
u
)
)
=
(
u
).
(7.80)
−
ψ(
1
u
Note that (
7.80
) is the expression we found earlier for the Laplace transform of the
memory kernel in terms of the waiting-time distribution density. Consequently we have
established the veracity of (
7.74
). This indirect method of establishing that (
7.74
)isthe
solution to the perturbed GME is actually much easier than solving the original equation
directly.
7.2.5
Stochastic resonance and conventional LRT
Stochastic resonance is a phenomenon that has attracted the attention of many investiga-
tors, too many to list here, since its introduction in the early 1980s [
14
,
30
] as a possible
explanation of the 100,000-year cycle in the climate record [
12
,
13
,
47
]. The intent was
to develop a simple model of the alterations of the Earth's climate between ice ages
and periods of warm weather due to the eccentricity of the Earth's orbit. It is currently
believed by the climatology community that this direct effect is not strong enough to
determine such a large change in climate. Benzi
et al
.[
12
,
13
] coincidently with Nico-
lis and Nicolis [
47
] conjectured that there may exist a cooperative mechanism between
the weak change of eccentricity in the Earth's orbit and the natural random fluctua-
tions in the average global temperature that might account for the periodic and strong
