INTRODUCTION xi

across a potential wall of height

v

2

is given to exponential order by T (ε) = exp(

v

ε

).

Hence, only in exponentially large scales of the form T (ε) = exp(

μ

ε

) parametrized by

an energy parameter μ we can expect to see effects of transitions between different

domains of attraction. We remark at this place that our approach essentially differs

from the one by Berglund and Gentz [13]. If b represents a negative potential

gradient for instance, their approach would typically not only scale time by T ,

but also the depths of the potential wells by a function of ε. As a consequence,

transitions even for the deterministic dynamical system become possible, and their

noise induced transitions happen on time scales of intermediate length. In contrast,

in our setting transitions between the domains of attraction of the deterministic

system are impossible, and noise induced ones are observed on very large time scales

of the order of Kramers’ time, typically as consequences of large deviations. The

function b is assumed to be one-periodic w.r.t. time, and so the system described

by (0.1) attains period T by re-scaling time in fractions of T . The deterministic

system

˙

ξ

t

= b(s, ξt) with frozen time parameter s is supposed to have two domains of

attraction that do not depend on s ≥ 0. In the “classical” case of a drift derived from

a potential, b(t, x) = −∇xU(t, x) for some potential function U, equation (0.1) is

analogous to the overdamped motion of a Brownian particle in a d-dimensional time

inhomogeneous double-well potential. In general, trajectories of the solutions of

differential equations of this type will exhibit randomly periodic behavior, reacting

to the periodic input forcing and eventually amplifying it. The problem of optimal

tuning at large periods T consists in finding a noise amplitude ε(T ) (the resonance

point) which supports this amplification effect in a best possible way. During the last

20 years, various concepts of measuring the quality of periodic tuning to provide

a criterion for optimality have been discussed and proposed in many applications

from a variety of branches of natural sciences (see Gammaitoni et al. [43] for an

overview). Its rigorous mathematical treatment was initiated only relatively late.

The first approach towards a mathematically precise understanding of stochas-

tic resonance was initiated by Freidlin [39]. To explain stochastic resonance in the

case of diffusions in potential landscapes with finitely many minima (in the more

general setting of (0.1), the potential is replaced by a quasi-potential related to the

action functional of the system), he goes as far as basic large deviations’ theory can

take. If noise intensity is ε, in the absence of periodic exterior forcing, the exponen-

tial order of times at which successive transitions between meta-stable states occur

corresponds to the work to be done against the potential gradient to leave a well

(Kramers’ time). In the presence of periodic forcing with period time scale e

μ

ε

, in

the limit ε → 0 transitions between the stable states with critical transition energy

close to μ will be periodically observed. Transitions with smaller critical energy

may happen, but are negligible in the limit. Those with larger critical energy are

forbidden. In case the two local minima of the potential have depths

V

2

and

v

2

,

v V , that switch periodically at time

1

2

(in scale T accordingly at time

T

2

), for

T larger than e

v

ε

the diffusion will be close to the deterministic periodic function

jumping between the locations of the deepest wells. As T exceeds this exponen-

tial order, many short excursions to the wrong well during one period may occur.

They will not count on the exponential scale, but trajectories will look less and

less periodic. It therefore becomes plausible that physicists’ quality measures for

periodic tuning which always feature some maximal tuning quality of the random