Environmental Engineering Reference
In-Depth Information
variable. The temporal dynamics are therefore modeled by the stochastic differential
equation d
φ/
d
t
=
f
(
φ
)
+
g
(
φ
)
ξ
(
t
). The functional usage of the DMN consists in
approximating the colored noise,
ξ
(
t
), as a DMN, i.e.,
ξ
(
t
)
=
ξ
dn
(
t
). In this case none
of the parameters
k
1
,
k
2
,
2
has an arbitrary value. In fact, these parameters
need to be determined by adapting the DMN to the characteristics of the driving noise
(i.e., for example, by matching the mean, variance, skewness, and correlation scale).
Moreover, the functions
f
(
1
,and
φ
φ
)and
g
(
) are in this case assigned a priori, whereas
f
1
(
φ
)and
f
2
(
φ
) are obtained from (
2.14
) and depend on the noise characteristics
f
1
(
(2.15)
To summarize, the functional or mechanistic usage of the dichotomous noise cor-
responds to two distinct approaches to the stochastic modeling of processes driven
by DMN. The differences may be relevant, in particular when dealing with noise-
induced transitions (see Chapter 3). Once the approach that is suitable for the study
of a specific problem is selected, the dichotomous noise provides a useful modeling
framework with a number of applications to the environmental sciences, as shown in
Chapter 4. Thus in the following subsection we present some probabilistic methods
to solve stochastic equations driven by dichotomous noise.
φ
)
=
f
(
φ
)
+
g
(
φ
)
1
,
f
2
(
φ
)
=
f
(
φ
)
+
g
(
φ
)
2
.
2.2.3 Stochastic processes driven by dichotomous noise
2.2.3.1 General framework
Consider the stochastic process
φ
(
t
) driven by multiplicative dichotomous noise,
d
d
t
=
f
(
φ
)
+
g
(
φ
)
ξ
dn
(
t
)
,
(2.16)
where
f
(
φ
)and
g
(
φ
) are two deterministic functions of the state variable
φ
.We
assume that both
f
(
φ
)and
g
(
φ
) are continuous for any value of
φ
. Depending on
the approach or interpretation used for the DMN, the functions
f
(
)are
assigned with a direct physical meaning (functional approach) or are obtainable from
functions
f
1
,
2
(
φ
)and
g
(
φ
) by use of Eqs. (
2.14
) (mechanistic approach).
Some examples can help us understand the role of the driving force in the dynamics
φ
of
(
t
).We consider four simple cases. The first three examples refer to themechanistic
usage of DMN, and the fourth case considers the functional usage:
Example 2.1:
φ
φ
(
t
) exponentially increases (decreases) when the noise is in the
1
(
2
)
state:
f
1
(
φ
)
=
1
−
φ,
f
2
(
φ
)
=−
φ.
(2.17)
An example is shown in Fig.
2.3
(a).
Example 2.2:
φ
(
t
) linearly increases (decreases) when the noise is in the
1
(
2
) state:
f
1
(
φ
)
=
1
,
f
2
(
φ
)
=−
1
.
(2.18)
An example of the resulting dynamics of
φ
(
t
) is shown in Fig.
2.3
(b).
Search WWH ::
Custom Search