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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