Noise-driven dynamical systems - Noise-induced Phenomena in the Environmental Sciences - page 28

p φ

2

1.75

1.5

1.25

1

0.75

0.5

0.25

φ

0.2 0.4 0.6 0.8 1

Figure 2.8. The pdfs corresponding to state-dependent (dashed curve, b =− 0 . 5,

θ 0 = 1 . 25) and constant (continuous curve, b = 0, θ 0 = 1 . 1) transition rates. The

common parameters are q ∗ = 1, α = 0 . 05, and σ = 0 . 1.

2.3 White shot noise
2.3.1 Definition and properties
WSN, or white Poisson noise, is a stochastic process described by a state variable
ξ τ i , with each pulse

having an infinitesimal duration and an infinite random height h i δ

sn ( t ) that is defined by a sequence of pulses at random times

(0), where δ ( · )is

the Dirac delta function (see Fig. 2.9 ). The random times { τ } form a Poisson sequence i

with rate λ , which implies that the probability distribution of the interarrival times

e − λ t t i = τ − τ 1 is p T ( t ) = λ (e.g., Ross , 1996 ). The probability distribution of i i −

1
α

δ α =

e − h /α . the random heights [divided by (0)] is exponential with mean , p H ( h )

A mathematical formulation for the process is

ξ sn ( t ) = h i δ ( t − τ i ) . (2.50)

i

Shot noise has been extensively investigated by scientists since the beginning of
the 20th century. The first documented works on shot noise were written by Campbell
( 1909a , 1909b ). A comprehensive analysis of shot noise was conducted by Rice ( 1944 ,
1945 ).
WSN is a singular mathematical object, just as the Dirac delta function is a singular
function. We can better understand these singularities by considering WSN as the
formal derivative of the homogeneous compound Poisson process

t
0 ξ sn ( t )d t =

Z ( t ) = h i ( t − τ i ) , (2.51)

i

where ) is the unit step function. A realization of the stochastic process Z ( t )is

showninFig. 2.9 and resembles a staircase with random steps.

( ·