6 1. HEURISTICS

The optimally tuned system is then said to be in stochastic resonance. Nicolis

[83] and Benzi et al. [5], by tuning the noise parameter ε to appropriate values, were

able to give qualitative explanations for glaciation cycles based on this phenomenon.

Stochastic resonance proved to be relevant in other elementary climate models

than the energy balance models considered so far. In Penland et al. [87], Wang

et al. [107, 106], a two-dimensional stochastic model for a qualitative explanation

of the ENSO (El Ni˜ no Southern Oscillation) phenomenon also leads to stochastic

resonance effects: for certain parameter ranges the model exhibits random tuned

transitions between two stable sea surface temperatures. New evidence for the pres-

ence of stochastic resonance phenomena in paleo-climatic time series was added by

Ganopolski and Rahmstorf [45]. Their paper interprets the GRIP ice core record

representing temperature proxies from the Greenland glacier that extend over a

period of roughly 90 000 years, and showing the fine structure of the temperature

record of the last glacial period. The time series shows about 20 intermediate warm-

ings during the last glacial period commonly known under the name of Dansgaard–

Oeschger events. These events are clearly marked by rapid spontaneous increases

of temperature by about 6K followed by slower coolings to return to the initial

basic cold age temperature. It was noted in [45] that a histogram of the number of

Dansgaard–Oeschger events with a duration of k · 1480 years, with k = 1, 2, 3,...

exhibits the typical shape of a stochastic resonance spike train consistent for in-

stance with the results of Herrmann and Imkeller [49] for Markov chains describing

the effective diffusion dynamics, or Berglund and Gentz [13] for diffusion processes

with periodic forcing.

1.2. Heuristics of our mathematical approach

The rigorous mathematical elaboration of the concept of stochastic resonance

is the main objective of this book. We start its mathematically sound treatment

by giving a heuristical outline of the main stream of ideas and arguments based on

the methods of large deviations for random dynamical systems in the framework of

the Freidlin–Wentzell theory. Freidlin [39] is able to formulate Kramers’ [65] very

old seminal approach mathematically rigorously in a very general setting, and this

way provides a lower estimate for the good tuning (see also the numerical results

by Milstein and Tretyakov [77]). To obtain an upper estimate, we finally argue by

embedding time discrete Markov chains into the diffusion processes that describe

the effective dynamics of noise induced transitions. Optimal tuning results obtained

for the Markov chains will then be transferred to the original diffusion processes.

To describe the idea of our approach, let us briefly return to our favorite ex-

ample explained in the preceding section. Recall that the function

f(t, T ) = Q(t) 1 − a(T ) − γ T

4,

T, t ≥ 0,

describes a multiple of Rin − Rout, and its very slow periodicity in t is initiated

by the assumption on the solar constant Q(t) = Q0 + b sin(ωt). Let us compare

this quantity, sketched in Figure 1.9 schematically for two times, say t1, t2 such

that Q takes its minimum at t1 and its maximum at t2. The graph of f moves

periodically between the two extreme positions. Note that in the one-dimensional

situation considered, f(t, ·) can be seen as the negative gradient of a potential

function U(t, ·) which depends periodically on time t.