Introduction
Speaking about noise we usually mean something that deteriorates the opera-
tion of a system. It is understood as a disturbance, a random and persistent one,
that obscures or reduces the clarity of a signal.
In nonlinear dynamical systems, however, noise may play a very constructive
role. It may enhance a system’s sensitivity to a small periodic deterministic signal
by amplifying it. The optimal amplification of small periodic signals by noise gives
rise to the ubiquitous phenomenon of stochastic resonance (SR) well studied in
a plethora of papers in particular in the physical and biological sciences. This
book presents a mathematical approach to stochastic resonance in a well defined
particular mathematical framework. We consider weakly periodic systems with
additive noise of small amplitude ε. The systems possess two domains of attraction
of stable equilibria separated by a manifold marking a barrier. Both the geometry
of the attraction domains as well as the barrier height are not subject to scalings
with the amplitude parameter ε. Therefore, as opposed to other approaches, noise
induced random transitions in our model happen on time scales of Kramers’ law,
i.e. they are exponential in the quotient of barrier height and noise amplitude,
and are due to large deviations. Our analysis is therefore based on a new large
deviations principle of the systems’ exit and transition dynamics between different
domains of attraction in the limit of small ε. It aims at the description of an
optimal interplay between large period length T of the weak periodic motion and
noise amplitude ε, where optimization is done with respect to appropriate measures
of quality of response of the stochastic system to the periodic input. We will
be uniquely concerned with the well founded and self contained presentation of
this mathematical approach mainly based on a space-time extension of Freidlin–
Wentzell’s theory of large deviations of noisy dynamical systems, first on a heuristic
and then on a mathematically rigorous level. The two principal messages of the
book are these. First we show that already in space dimension one the
classical physical measures of quality of periodic tuning such as the spectral power
amplification, due to the phenomenon of the small oscillations catastrophe, are not
robust with respect to dimension reduction. Comparing optimal tuning rates for
the diffusion processes and the finite state Markov chains retaining the models’
essentials one gets essentially different results (Chapter 3, Theorems 3.50, 3.53).
We therefore propose in arbitrary finite space dimension measures of quality
of periodic tuning based uniquely on the transition dynamics and show that these
measures are robust and, via a crucial large deviations result, are able to explain
stochastic resonance as optimal tuning (Chapter 4, Theorems 4.19, 4.29, 4.31).
Concentrating on these more theoretical themes, the book sheds some light on the
mathematical shortcomings and strengths of different concepts used in theory and
application of stochastic resonance, in a well defined framework. It does not aim at
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