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

2π

pq

V − v

v

exp

V + v

2ε

.

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