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 ⎪π−(0) ⎪π+(0) T = ϕ ϕ + ψ + ψ ϕ + ψ · (1 ϕ ψ) T 2 1 + (1 ϕ ψ) T 2 , T = 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) = (Q∗)lπT 1 (0) for 0 l T 2 1, and πT (l) = (Q∗)l(Q∗) 2 1 T 2 πT (0) for T 2 l T 1 to obtain (1.13). Note also the symmetry π−(l T + T 2 ) = π+(l) T and π+(l T + T 2 ) = π−(l), T 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 ) = T ξT (1) 2 , ε 0,T 2N. Here T 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 T ξT (1) = 1 T T −1 k=0 T Y ε (k)e 2πi T k = 1 eπi T T 2 −1 k=0 (π+(k) T π−(k))e T 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 ) · ψ)2 + 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
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