xii INTRODUCTION

trajectories to the periodic input signal cannot be captured by this phenomenon of

quasi-deterministic periodicity at very large time scales.

These quality measures, studied in Pavlyukevich [86] and Imkeller and Pavlyuke-

vich [59] assess quality of tuning of the stochastic output to the periodic determin-

istic input. The concepts are mostly based on comparisons of trajectories of the

noisy system and the deterministic periodic curve describing the location of the

relevant meta-stable states, averaged with respect to the equilibrium measure (of

the diffusion as a space-time process with time component given by uniform motion

in the period interval). Again in the simple one-dimensional situation considered

above the system switches between a double well potential state U with two wells of

depths

V

2

and

v

2

, v V, during the first half period, and the spatially opposite one

U(·) for the second half period. If as always time is re-scaled by T , the total period

length is T , and stochastic perturbation comes from the coupling to a white noise

of intensity ε. The most important measures of quality studied are the spectral

power amplification and the related signal-to-noise ratio, both playing an eminent

role in the physical literature (see Gammaitoni et al. [43], Freund et al. [41]). They

mainly contain the mean square average in equilibrium of the Fourier component

of the solution trajectories corresponding to the input period T , normalized in dif-

ferent ways. These measures of quality are functions of ε and T , and the problem

of finding the resonance point consists in optimizing them in ε for fixed (large) T .

Due to the high complexity of original systems, when calculating the resonance

point at optimal noise intensity, physicists usually pass to an effective dynamics

description. It is given by a simple caricature of the system reducing the diffu-

sion dynamics to the pure inter well motion (see e.g. McNamara and Wiesenfeld

[74]). The reduced dynamics is represented by a continuous time two state Markov

chain with transition probabilities corresponding to the inverses of the diffusions’

Kramers’ times. One then determines the optimal tuning parameters ε(T ) for large

T for the approximating Markov chains in equilibrium, a rather simple task. To

see that the Markov chain’s behavior approaches the diffusion’s in the small noise

limit, spectral theory for the infinitesimal generator is used. The latter is seen to

possess a spectral gap between the second and third eigenvalues, and therefore the

closeness of equilibrium measures can be well controlled. Surprisingly, due to the

importance of small intra well fluctuations, the tuning and resonance pattern of

the Markov chain model may differ dramatically from the resonance picture of the

diffusion. Subtle dependencies on the geometrical fine structure of the potential

function U in the minima beyond the expected curvature properties lead to quite

unexpected counterintuitive effects. For example, a subtle drag away from the other

well caused by the sign of the third derivative of U in the deep well suﬃces to make

the spectral power amplification curve strictly increasing in the parameter range

where the approximating Markov chain has its resonance point.

It was this lack of robustness against model reduction which motivated Her-

rmann and Imkeller [50] to look for different measures of quality of periodic tun-

ing for diffusion trajectories. These notions are designed to depend only on the

rough inter well motion of the diffusion. The measure treated in the setting of

one-dimensional diffusion processes subject to periodic forcing of small frequency

is related to the transition probability during a fixed time window of exponential

length T (ε) = exp(

μ

ε

) parametrized by a free energy parameter μ according to the