14 1. HEURISTICS

• the total energy of the averaged trajectories

EnX

(ε, T ) =

1

0

Eν XsT

2

ds, ε 0, T 0.

The second family of criteria is more probabilistic. It refers to quality measures

purely based on the location of transition times between domains of attraction of

the local minima, and residence time distributions measuring the time spent in one

well between two transitions, or interspike times. This family, to be discussed in

more detail in Section 1.4 below is certainly less popular in the physics community.

As will turn out later, these physical notions of quality of periodic tuning

of random trajectories exhibit one important drawback: they are not robust with

respect to model resolution. It is here that an important concept of model reduction

enters the stage. It is based on the conjecture that the effective dynamical properties

of periodically forced diffusion processes as given by (1.3) can be traced back to

finite state Markov chains periodically hopping between the stable equilibria of

the potential function underlying the diffusion, for which the smallness parameter

of the noise intensity is simply reflected in the transition matrix. These Markov

chains should be designed to capture the essential information about the inter-well

dynamics of the diffusion, while intra-well small fluctuations of the diffusion near

the potential minima are neglected. Investigating goodness of tuning according to

the different physical measures of quality makes sense both for the Markov chains

as for the diffusions. This idea of model reduction was captured and followed in the

physics literature in Eckmann and Thomas [32], McNamara and Wiesenfeld [74],

and Nicolis [83]. In fact, theoretical work on the concept of stochastic resonance

in the physics literature is based on the model reduction approach, see the surveys

Anishchenko et al. [1], Gammaitoni et al. [43, 44], Moss et al. [79], and Wellens

et al. [108].

As we shall see in Chapter 3, the optimal tuning relations between ε and T

do not necessarily agree for Markov chains and diffusions. Even in the small noise

limit discrepancies may persist that are caused by very subtle geometric properties

of the potential function. It is our goal to present a notion of quality of periodic

tuning which possesses this robustness property when passing from the Markov

chains capturing the effective dynamics to the original diffusions. For this reason

we shall study the different physical notions of quality of tuning first in the context

of typical finite state Markov chains with periodically forced transition matrices.

1.3. Markov chains for the effective dynamics and the physical

paradigm of spectral power amplification

To keep this heuristic exposition of the main ideas of our mathematical ap-

proach as simple as possible, besides allowing only two states for our Markov chain

that play the role of the stable equilibria of the potential −1 and 1, let us also dis-

cretize time. We continue to assume as in the discussion of periodically switching

potential states above that U1(−1) = U2(1) = −

V

2

, and U1(1) = U2(−1) = −

v

2

.

In a setting better adapted to our continuous time diffusion processes, in Chapter

3 time continuous Markov chains switching between two states will capture the

effective diffusion dynamics. Hence, we follow here Pavlyukevich [86] and Imkeller

and Pavlyukevich [58] and shall assume in this section that the parameter T in

our model describing the period length, is an even integer. So for T ∈ 2N, ε 0,