INTRODUCTION xiii

Kramers–Eyring formula. The corresponding exit rate is maximized in μ to ac-

count for optimal tuning. The methods of investigation of stochastic resonance in

[50] are heavily based on comparison arguments which are not an appropriate tool

from dimension 2 on. Time inhomogeneous diffusion processes such as the ones

under consideration are compared to piecewise homogeneous diffusions by freezing

the potential’s time dependence on small intervals.

In Herrmann et al. [51] this approach is extended to the general setting of finite

dimensional diffusion processes with two meta-stable states. Since the stochastic

resonance criterion considered in [50] is based on transition times between them,

our analysis relies on a suitable notion of transition or exit time parametrized again

by the free energy parameter μ from T (ε) = exp(

μ

ε

) as a natural measure of scale.

Assume now that the depths of the two equilibria of the potential in analogy to

the scenarios considered before are smooth periodic functions of time of period 1

given for one of them by

v(t)

2

, and for the other one by the same function with some

phase delay (for instance by

1

2

). Therefore, at time s the system needs energy v(s)

to leave the domain of attraction of the equilibrium. Hence an exit from this set

should occur at time

aμ = inf{t ≥ 0: v(t) ≤ μ}

in the diffusion’s natural time scale, in the time re-scaled by T (ε) thus at time

aμ · T (ε). To find a quality measure of periodic tuning depending only on the

transition dynamics, we look at the probabilities of transitions to the other domain

within a time window [(aμ − h)T (ε), (aμ + h)T (ε)] centered at aμ · T (ε) for small

h 0. If τ is the random time at which the diffusion roughly reaches the other

domain of attraction (to be precise, one has to look at first entrance times of small

neighborhoods of the corresponding equilibrium), we use the quantity (again, to be

precise, we use the worst case probability for the diffusion starting in a point of a

small neighborhood of the equilibrium of the starting domain)

Mh(ε,

μ) = P τ ∈ [(aμ − h)T (ε), (aμ + h)T (ε)] .

To symmetrize this quality measure with respect to switching of the equilibria, we

refine it by taking its minimum with the analogous expression for interchanged

equilibria. In order to exclude trivial or chaotic transition behavior, the scale

parameter μ has to be restricted to an interval IR of reasonable values which we

call resonance interval. With this measure of quality, the stochastic resonance point

may be determined as follows. We first fix ε and the window width parameter h 0,

and maximize

Mh(ε,

μ) in μ, eventually reached for the time scale μ0(h). Then the

eventually existing limit limh→0 μ0(h) will be the resonance point.

To calculate μ0(h) for fixed positive h we use large deviations techniques. In

fact, our main result consists in an extension of the Freidlin–Wentzell large devia-

tions result to weakly time inhomogeneous dynamical systems perturbed by small

Gaussian noise which states that

lim

ε→0

ε ln 1 −

Mh(ε,

μ) = μ − v(aμ − h),

again in a form which is symmetric for switched equilibria. We show that this

asymptotic relation holds uniformly w.r.t. μ on compact subsets of IR, a fact which

enables us to perform a maximization and find μ0(h). The resulting notion of

stochastic resonance is strongly related to the notions of periodic tuning based

on interspike intervals (see [49]), which describe the probability distribution for