1.2. HEURISTICS OF OUR MATHEMATICAL APPROACH 13

(1.8), Freidlin in [39, Theorem 1] proves that independently of the starting point

x it should stay near −1 in the following sense. The Lebesgue measure of the set

t ∈ [0, 1]: |XtT

ε

(ε)

− (−1)| δ

converges to 0 in probability as ε → 0. If T (ε) e

V

s

ε

, the time left is not long

enough for any crossing: the particle, starting at x, stays in the starting well, near

the stable equilibrium point. In other words, the Lebesgue measure of the set

t ∈ [0, 1]: |XtT

ε

(ε)

− (−I(−∞,0)(XtT

ε

(ε)

) + I[0,∞)(XtT

ε

(ε)

))| δ

converges to 0 in the small noise limit. This observation is at the basis of Freidlin’s

law of quasi-deterministic periodic motion discussed in the subsequent section. The

lesson it teaches is this: to observe switching of the position to the energetically

most favorable well, T (ε) should be larger than some critical level e

λ

ε

, where λ =

infs≥0 V

s

. Measuring time in exponential scales by μ through the equation T (ε) =

e

μ

ε

, the condition translates into μ λ. Continuing the reasoning of the preceding

subsection, if this condition is satisfied, we may define a periodic function φ denoting

the deepest well position in dependence on t. Then, in generalization of (1.11), the

process

Xε

will spend most of the time, measured by Lebesgue’s measure, near φ

for small ε.

1.2.5. Quality of periodic tuning and reduced motion. Do the mani-

festations of quasi-deterministic motion in instantaneously frozen potentials just

discussed explain stochastic resonance? The problem is obvious. They just give

lower bounds for the scale T (ε) = e

μ

ε

for which noise strength ε leads to random

switches between the most probable potential wells near the (periodic) deterministic

times when the role of the deepest well switches. But if μ is too big, occasional

excursions into the higher well will destroy a truly periodic tuning with the po-

tential (see Figure 1.12). Just the duration of the excursions, being exponentially

smaller than the periods of dwelling in the deeper well, will not be noticed by

the residence time criteria discussed. We therefore also need an upper bound for

possible scales. In order to find this optimal tuning scale μR λ, we first have

to measure goodness of periodic tuning of the trajectories of the solution. In the

huge physics literature on stochastic resonance, two families of criteria can be dis-

tinguished. The first one is based on invariant measures and spectral properties of

the infinitesimal generator associated with the diffusion Xε. Now, Xε is not time

autonomous and consequently does not admit invariant measures. By taking into

account deterministic motion of time in the interval of periodicity and considering

the time autonomous process Zt

ε

= (t mod T (ε),Xt

ε),

t ≥ 0, we obtain a Markov

process with an invariant measure νt

ε(x)

dt dx. In particular, for t ≥ 0 the law of

Xt

ε

∼ νt

ε(x)

dx and the law of Xt+T

ε

(ε)

∼ νt+T

ε

(ε)

(x) dx, under this measure are the

same for all t ≥ 0. Let us present the most important notions of quality of tuning

(see Jung [62], or Gammaitoni et al. [43]):

• the spectral power amplification (SPA) which plays an eminent role in

the physics literature and describes the energy carried by the spectral

component of the averaged trajectories of

Xε

corresponding to the period

of the signal:

ηX

(ε, T ) =

1

0

Eν XsT

ε

·

e2πis

ds

2

, ε 0, T 0.