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
, 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(
) parametrized by a free energy parameter μ according to the
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