Environmental Engineering Reference
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manipulations of the general expressions provided by Hanggi [ 1978 , Eq. (2.15),
p. 409]:
i ! g ( i ) (
) ,
κ i
ξ =
g (
φ
)
φ
(B3.2-3)
i
=
1
κ i is the cumulant of the i th order of the increments of the integrated process
where
= t
z ( t )
0 ξ
( t )d t (see van Kampen , 1992 , p. 238), g (1) (
φ
)
=
g (
φ
), and
) g ( i ) (
g ( i + 1) (
) (the same functions were used in Subsection 2.3.4 ). When the
noise is white and Gaussian, the integrated process z ( t ) is the Wiener process, [see
Eq. ( 2.75 )], and
φ
)
=
g (
φ
φ
κ i =
0 for any i , except
κ 2 =
2 s gn ;Eq.( B3.2-2 ) is therefore easily
recovered from Eq. ( B3.2-3 ).
The corresponding integrated process for shot noise is instead the compound Poisson
process in Eq. ( 2.51 ), and
i . We therefore obtain
κ i = λ
i !
α
i g ( i ) (
) .
g (
φ
)
ξ sn = λ
i = 1 α
φ
(B3.2-4)
If g (
φ
) is a linear function of
φ
, g (
φ
)
=
a
+
b
φ
(where a and b are two coefficients),
Eq. ( B3.2-4 ) further specifies
a
+
b
φ
g (
φ
)
ξ sn =
( a
+
b
φ
)
ξ sn = αλ
,
(B3.2-5)
1
b
α
which is valid for b
1 (otherwise the process diverges). It is clear from Eqs. ( B3.2-4 )
and ( B3.2-5 ) that the shot noise (when its effect in the Langevin equation is interpreted
in the Stratonovich sense) may also play an important role in inducing phase transitions
in the dynamics of the system.
To generalize the notation we introduce a function g S (
α<
φ
) of the state variable and
write
, where s m is the intensity of the generic multiplicative
noise. It is clear that the specific function g S (
g (
φ
)
ξ m =
s m
g S (
φ
)
φ
) varies depending on the type of noise
and its interpretation. g S (
const, or the noise term in the Langevin
equation is interpreted in Ito's sense. In the case of Langevin equations with Gaussian
white noise, interpreted in the Stratonovich sense, we have g S (
φ
)
=
0 when g (
φ
)
=
) g (
φ
)
=
g (
φ
φ
).
ξ
β>
where
gn ( t ) is Gaussian white noise with intensity s gn .When
0 the domain
, +∞
boundaries are [0
[ and the steady-state pdf is (under Stratonovich's interpreta-
tion)
s β/ s gn
β
s gn
φ
s gn
gn
1
e
s gn φ
p (
φ
)
=
.
(3.51)
The pdf in Eq. ( 3.51 ) can be recognized to be a gamma distribution, whose modes
and antimodes are
s gn when
s gn ;
φ
=
0
,
φ
= β
β>
(3.52)
m
,
1
m
,
2
 
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