Environmental Engineering Reference
In-Depth Information
2.5
φ
t
2.5
φ
t
2
2
1.5
1.5
1
1
0.5
0.5
t
20
t
5
10
15
20
5
10
15
(a)
(b)
Figure 2.11. Examples of trajectories of (a) process (
2.66
) and (b) process (
2.67
). In
this simulation
α
=
0
.
5 and
λ
=
1.
hence it is incompatible with the ordinary rules of calculus. We show in Section 2.4
that convention (
2.63
) is instead appropriate in the case of Gaussian white noise.
2.3.4.1 Some examples of processes driven by WSN
To facilitate the understanding of the properties presented in the following sections,
it is useful to introduce some simple examples of processes driven by WSN:
Example 2.5:
φ
(
t
) decreases hyperbolically during the interarrival times and the noise is
additive:
d
d
t
=−
φ
2
2
+
αλ
+
ξ
sn
.
+
ξ
sn
=−
φ
(2.66)
Example 2.6:
The deterministic component is the same as in the previous example, but the
noise is multiplicative with
g
(
φ
)
=
φ
:
d
d
t
=−
φ
+
φξ
sn
.
2
+
φξ
sn
=−
φ
(
φ
−
αλ
)
(2.67)
(
t
) for these two processes.
In the second example [Fig. 2.11(b)], the size of the jumps
Figure
2.11
shows examples of the trajectories of
φ
in the trajectories of
φ
is evaluated with Eq. (
2.60
), which in this simple case [with
g
(
φ
)
=
φ
] can be
analytically integrated. We obtain
=
φ
1
−
φ
0
=
φ
0
(
e
h
−
1)
,
(2.68)
with a clear dependence of
on the spike intensity
h
and the starting point
φ
0
. Thus the
function
g
(
) modulates the magnitude of the jumps. In particular, the multiplicative
character of the noise implies that if
φ
(
e
h
φ
0
is greater (lower) than
h
/
−
1) the jump
size in the
trajectory is greater (lower) than
h
. The differences of the jump sizes in
the two examples can be observed in Fig.
2.11
, for which the same realization of the
shot noise
φ
ξ
sn
is used.
2.3.4.2 Steady-state pdf and its properties
Under the Stratonovich interpretation, we obtain the steady-state pdf for a process
driven by WSN from (
2.31
) by taking the limits in (
2.56
)(butwith
=−
λα
to have
2
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