Environmental Engineering Reference
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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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