Noise-driven dynamical systems - Noise-induced Phenomena in the Environmental Sciences - page 10

Box 2.1: Transient dynamics of the dichotomous Markov process

Some considerations of the transient dynamics of the dichotomous Markov process can
be of interest. Equations ( 2.2 ) and ( 2.3 ) can be solved to give

= τ c k 2 1 e − t /τ c +

P 1 (0) e − t /τ c

P 1 ( t ) − , (B2.1-1)

= τ c k 1 1 e − t /τ c + P 2 (0) e − t /τ c P 2 ( t ) − , (B2.1-2)

where P 1 (0) and P 2 (0) = 1 − P 1 (0) are the initial conditions and

1
k 1 +

τ 1 τ 2
τ 1 + τ 2

τ c =

k 2 =

(B2.1-3)

is the characteristic relaxation time of the process.
It can be argued from Eqs. ( B2.1-1 ) and ( B2.1-2 ) that the joint probability for ξ dn at

time t and t is

prob ξ dn ( t ) = j = i ,ξ dn ( t ) p i , j =

c 1

τ c

e − | t − t |

2

= (1 − k i τ c )(1 − k j τ c ) τ −

k i τ c ) e − | t − t | + δ i , j (1 − , (B2.1-4)

τ c

where δ i , j is the Kronecker delta function and i , j = 1 , 2.

The steady-state autocorrelation function is then

2

2 k 1 ) e − | t − t |

ξ dn ( t )

2
1 k 2 +

2

ξ dn ( t ) = p i , j i j = τ c (

τ c

i , j = 1

s dn
τ c

=− 1 2 e − | t − t | e − | t − t |

= , (B2.1-5)

τ c τ c

where Eq. ( 2.8 ) has been repeatedly used. The term

3
c (

2 − 1 ) 2 s dn = k 1 k 2 τ =− 1 2 τ c (B2.1-6)

in Eq. ( B2.1-5 ) represents the noise amplitude or intensity .

integral scale is generally interpreted as a measure of the memory of the process, and
in the case of dichotomous noise it is

1
k 1 +

I =

k 2 = τ c .

(2.11)

Some generalization of dichotomous noise were proposed in the literature. Notable
examples include the so-called trichotomous noise ( Mankin et al. , 1999 ), characterized
by a three-valued state space and its further generalization, multivalued noise ( We i ss
et al. , 1987 ); compound dichotomous noise ( van den Broeck , 1983 ), in which the value