Noise-driven dynamical systems - Noise-induced Phenomena in the Environmental Sciences - page 41

periods. The intermittent nature of a rainfall time series is lost when the temporal
resolution (or aggregation scale) of rainfall is much larger than the diurnal one. For
example, if we consider rainfall at an annual resolution, we typically find a sequence
of random uncorrelated precipitation amounts, approximatively Gaussian distributed.
If the response time of the dynamical system subjected to precipitation is larger than
the year (e.g., slowly growing vegetation or geomorphological systems), the forcing
can therefore be modeled as white Gaussian noise. Other examples of environmental
processes in which the forcing can be represented as white Gaussian noise are given
in Chapters 4 and 6.

2.4.4 Stochastic process driven by Gaussian noise
In this case the stochastic equation regulating the dynamics of the state variable φ is

d d t =

f ( φ ) + g ( φ ) ξ gn . (2.80)

As in the case of a system driven by WSN, a problem of interpretation arises when
the noise is multiplicative [i.e., g (

φ φ

]. We follow a similar reasoning
as in Subsection 2.3.4 and adopt a Stratonovich convention to interpret Eq. ( 2.80 ).
In the case of systems forced by Gaussian white noise, the Stratonovich convention
corresponds to calculating

) is a function of

before and
after the jump; see Eq. ( 2.63 ). In fact, in this case only the terms up to h 2 remain in
Eqs. ( 2.62 )and( 2.65 ), because h is infinitesimal, of the order of

φ in g ( φ )ashalfthesumofthevaluesof φ

∞ − 0 . 5 [see relations
( 2.79 )]. As a consequence, Eqs. ( 2.62 )and( 2.65 ) are identical for systems driven
by Gaussian white noise, i.e., the Stratonovich rule (which we have broadly defined
as the rule that arises from the use of the ordinary rules of calculus) corresponds to
calculating g ( φ 1 are the values immediately before

and after the jump. We remark again that this is valid only for systems driven by
Gaussian white noise.

φ )in( φ 0 + φ 1 ) / 2, where φ 0 and

2.4.4.1 Some examples of processes driven by Gaussian white noise
To facilitate the understanding of the properties presented in following subsections, it
is useful to introduce some simple examples of processes driven by Gaussian white
noise.
Example 2.7: f (

φ =− φ

3

+

) 2 and the noise is additive:

d d t =− φ

3

+ 2 + ξ gn . (2.81)

Example 2.8: The deterministic component is the same as in the previous example, but the
noise is multiplicative with g ( φ ) = φ :

d d t =− φ

3

+ 2 + φξ gn . (2.82)