18 1. HEURISTICS
the amplification coefficient
ηY
(ε, T ) as a function of ε. So the stochastic resonance
point is marked by the maximum of the spectral power amplification coefficient as
a function of ε. To calculate it, substitute
e−1/ε
= x, and differentiate the explicit
formula (1.14). The resulting relationship between period length T (ε) and noise
intensity ε marking the stochastic resonance point can be recast in the formula
T (ε)

=
1

pq
V v
v
exp
V + v

.
The maximal value of spectral power amplification is given by
lim
ε→0
ηY
(ε, T (ε)) =
4
π2
0
ε
1 2
ηY (ε, T )
4
π 2
Figure 1.13. The coefficient of the spectral power amplification
ε ηY (ε, T ) for p = q = 0.5, V = 2, v = 1, T = 10 000.
We also see that the spectral power amplification as a measure of quality of sto-
chastic resonance allows to distinguish a unique time scale, and find its exponential
rate
V +v
2
together with the pre-exponential factor. The optimal exponential rate
is therefore given by the arithmetic mean of the two potential barriers marked by
the deep and shallow well of our double well potential. This basic relationship will
appear repeatedly at different stages of our mathematical elaboration of concepts
of optimal tuning.
We may summarize our findings so far for discrete Markov chains that capture
the effective dynamics of the potential diffusions which are our main subject of in-
terest. Following the physics literature (e.g. Gammaitoni et al. [43] and McNamara
and Wiesenfeld [74]) we understand stochastic resonance as optimal spectral power
amplification. The closely related notion of signal-to-noise ratio and other reason-
able concepts based on quality measures such as the relative entropy of invariant
laws are discussed for Markov chains in Chapter 3 (see Section 3.2). The spectral
power amplification coefficient measures the power carried by the expected Fourier
coefficient in equilibrium of the Markov chain switching between the stable equilib-
ria of the potential landscape of the diffusion which corresponds to the frequency
of the underlying periodic deterministic signal.
1.4. Diffusions with continuously varying potentials
The concept of spectral power amplification is readily extended to Markov
chains in continuous time, still designed to capture the effective diffusion dynamics
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