Noise-driven dynamical systems - Noise-induced Phenomena in the Environmental Sciences - page 50

The pdfs are calculated with relation ( 2.106 ), which is valid [see condition ( 2.107 )]
for

2 )

1 − τ ou (1 − 3 φ > 0 ⇒ τ < 1 . (2.109) ou

Two noise intensities and different correlation scales are shown. The figure clearly
shows that in this case the effect of noise correlation cannot be neglected. In fact,
remarkable differences exist with respect to the white-noise case (i.e.,

τ =

0), in- ou

cluding the emergence of sharper peaks in p ( ) and the (weak) shift of the modes

with increasing values of the correlation scale. In Chapters 3 and 4 we show how the
autocorrelation of noise is also able to induce more structural effects on the shape of
p (

φ

).
To show and compare the effect of the different approximations presented in this
section, we calculated p (

φ

) for the same model ( 2.108 ) by using Eqs. ( 2.103 ), ( 2.104 ),
and ( 2.106 ). In this case the three expressions for p (

φ

φ ) give very similar results (see

Fig. 2.17 ).