1.3. EFFECTIVE DYNAMICS VIA DICRETE MARKOV CHAINS 17

Using some more elementary algebra we find

(Q2)

∗

T

2

(Q1)

∗

T

2

= (Q1)

T

2

(Q2)

T

2

∗

=

1 − ψ ψ

ϕ 1 − ϕ

T

2

1 − ϕ ϕ

ψ 1 − ψ

T

2

=

1

ϕ + ψ

ϕ ϕ

ψ ψ

+ (1 − ϕ − ψ)

T

2

ϕ − ψ

ϕ + ψ

−1 −1

1 1

+

(1 − ϕ −

ψ)T

ϕ + ψ

ϕ −ψ

−ϕ ψ

,

from which another straightforward calculation yields

⎧

⎪

⎪

⎪πT

⎨

⎪

⎪

⎪πT

⎩

−(0)

=

ϕ

ϕ + ψ

+

ψ

ϕ + ψ

·

(1 − ϕ − ψ)

T

2

1 + (1 − ϕ − ψ)

T

2

,

+(0)

=

ψ

ϕ + ψ

+

ϕ

ϕ + ψ

·

(1 − ϕ − ψ)

T

2

1 + (1 − ϕ − ψ)

T

2

.

To compute the remaining entries, we use πT (l) =

(Q1)lπT ∗

(0) for 0 ≤ l ≤

T

2

− 1,

and πT (l) = (Q2)l(Q1) ∗ ∗

T

2

πT (0) for

T

2

≤ l ≤ T − 1 to obtain (1.13). Note also the

symmetry πT

−(l

+

T

2

) = πT

+(l)

and πT

+(l

+

T

2

) = πT

−(l),

0 ≤ l ≤

T

2

− 1.

To motivate the physical quality of tuning concept of spectral power amplifica-

tion, we first remark that our Markov chain Y ε can be interpreted as amplifier of

the periodic input signal of period T . In the stationary regime, i.e. if the law of

Y ε is given by the measure πT , the power carried by the output Markov chain at

frequency a/T is a random variable

ξT (a) =

1

T

T −1

l=0

Y

ε(l)e

2πia

T

l.

We define the spectral power amplification (SPA) as the relative expected power

carried by the component of the output with (input) frequency

1

T

. It is given by

ηY

(ε, T ) = EπT ξT (1)

2

, ε 0,T ∈ 2N.

Here EπT denotes expectation w.r.t. the stationary law πT .

The explicit description of the invariant measure now readily yields an explicit

formula for the spectral power amplification. In fact, using (1.13) one immediately

gets

EπT ξT (1) =

1

T

T −1

k=0

EπT Y

ε(k)e 2πi

T

k

=

1 −

eπi

T

T

2

−1

k=0

(πT

+(k)

− πT

−(k))e 2πi

T

k

=

4

T

ϕ − ψ

ϕ + ψ

1

1 − e

2πi

T

−

1

1 − (1 − ϕ − ψ)e

2πi

T

.

Elementary algebra then leads to the following description of the spectral power

amplification coeﬃcient of the Markov chain Y

ε

for ε 0, T ∈ 2N:

(1.14)

ηY

(ε, T ) =

4

T 2

sin2( π

T

)

·

(ϕ −

ψ)2

(ϕ + ψ)2 + 4(1 − ϕ − ψ)

sin2( π

T

)

.

Note now that the one-step probabilities Q1 and Q2 depend on the parameters

noise level ε. Our next goal is to tune this parameter to a value which maximizes