Environmental Engineering Reference
In-Depth Information
p
φ
p
φ
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
φ
φ
1
0
1
2
3
1
2
3
4
(a)
(b)
Figure 2.15. Examples of steady-state pdfs for processes driven by Gaussian white
noise, corresponding to (a) Eq. (
2.81
) and (b) Eq. (
2.82
). The continuous curves
correspond to a noise intensity
s
gn
=
1, the dashed curves to
s
gn
=
2.
The pdfs in Eqs. (
2.85
)and(
2.86
) are shown in Figs.
2.15
(a) and 2.15(b), respec-
tively.
2.5 Colored Gaussian noise
In the introduction we discussed the fact that when the integral scale
τ
n
of the noise
and the typical time scale
τ
s
of the modeled dynamical system are comparable, the
noise correlation plays a fundamental role in the stochastic dynamics of the state
variable
and cannot be neglected. In spite of the existence of a plethora of different
types of correlated noises, exact analytical results are known only for some classes
of systems driven by non-Markovian Gaussian colored noises and for two types of
Markovian colored noise, namely the dichotomous noise and the Markovian Gaussian
colored noise, i.e., the Ornstein-Uhlenbeck (O-U) process. The case of dichotomous
noise was presented at the beginning of this chapter (Section
2.2
); in this section we
describe some basic properties of Gaussian colored noise.
We consider the stochastic model
d
φ
d
t
=
f
(
φ
)
+
g
(
φ
)
ξ
cn
(
t
)
,
(2.87)
driven by generic (Markovian or non-Markovian) Gaussian colored noise
ξ
cn
(
t
), with
zero mean [i.e.,
ξ
cn
(
t
)
=
0] and autocovariance function
ξ
cn
(
t
)
t
)
t
,
ξ
cn
(
t
)
=
C
(
t
−
,
t
≥
(2.88)
t
) is a generic function of
t
t
. The pdf
p
(
where
C
(
t
−
−
φ,
t
) can be formally written
as
p
(
φ,
t
)
=
δ
(
φ
(
t
)
−
φ
)
,
(2.89)
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