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
