Environmental Engineering Reference

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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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