Environmental Engineering Reference
InDepth Information
and
a
t
0
e
b
(
t
−
t
)
d
t
.
=
φ
0
e
bt
β
(
t
,φ
0
)
+
(2.97)
2.5.2 Dynamics driven by the OrnsteinUhlenbeck process
In the case of nonlinear Langevin equations, the solution of master equation (
2.90
)
remains a rather intractable problem, unless the noise is assumed to be Markovian.
In this case some important approximated results can be obtained. It can be shown
(Doob's theorem;
Doob
,
1942
) that there is only one form of Markovian colored
Gaussian noise. This noise is known as the OrnsteinUhlenbeck (OU) process (the
OU process is therefore the only existing Markovian, Gaussian colored process) and
is described by the equation
√
s
ou
τ
d
ξ
ou
d
t
1
τ
=−
ou
ξ
+
ou
ξ
gn
(
t
)
,
(2.98)
ou
where
τ
ou
is the correlation scale of the OU process,
s
ou
is the intensity of
ξ
ou
,and
ξ
gn
gn
(
t
)
t
)]. The
is Gaussian white noise with strength
s
gn
=
1 [i.e.,
ξ
gn
(
t
)
ξ
=
2
δ
(
t
−
autocorrelation function of the OU process is exponential:
s
ou
τ
ou
e
−

t
−
t

t
)
C
(
t
−
=
.
(2.99)
τ
ou
Notice that, although the OU process is Markovian, a process driven by OU
noise is not Markovian. Unfortunately, no exact mathematical expressions exist for
the steadystate pdf
p
(
φ
) associated with stochastic dynamical system (
2.87
)driven
by an OU process [i.e., with driving colored noise
ξ
ξ
ou
(
t
)].
However, some approximated solutions were proposed in the literature. Essentially,
two approaches were followed. The first one is to enlarge the state space, including
ξ
ou
(
t
) as an auxiliary variable driven by Gaussian white noise. In this way, the (
cn
(
t
) corresponding to
φ,ξ
ou
)
dynamics are a bivariate Markovian process and an exact twodimensional (2D)
FokkerPlanck equation can be written for the density function
p
(
t
). From
this starting point, several techniques were proposed to investigate the process
φ,ξ
ou
,
φ
(
t
),
including a perturbative expansion in
τ
ou
(
Horsthemke and Lefever
,
1984
) or continued
fraction expansions (
Risken
,
1984
). The other approach treats the process as driven
by nonMarkovian Gaussian noise and tries to obtain from exact master equation
(
2.90
) a FokkerPlancklike master equation. This equation is assumed to be able
to capture some characteristics of the nonMarkovian process
(
t
), at least in the
steadystate condition. The differences between these two approaches translate into
different ranges of validity of the approximated expressions, particularly with respect
to the correlation scale of the noise. Here we summarize the main results along with
their range of validity.
φ
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