Preface

Stochastic resonance is a phenomenon arising in many systems in the sciences

in a wide spectrum extending from physics through neuroscience to chemistry and

biology. It has attracted an overwhelming attendance in the science literature for

the last two decades, more recently also in the mathematics literature. It is generally

understood as the optimal amplification of a weak periodic signal in a dynamical

system by random noise.

This book presents a mathematical approach of stochastic resonance in a well

defined framework. We consider weakly periodic systems in arbitrary finite di-

mension with additive noise of small amplitude ε. They 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 scaled

with the amplitude parameter ε. Therefore, in contrast to other approaches, noise

induced random transitions in our model happen on time scales given by the expo-

nential of the quotient of barrier height and noise amplitude (Kramers’ times), and

are due to large deviations. Our analysis is therefore based on a new space-time

large deviations principle for the system’s 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. Optimization is done with respect to appropriate measures

of quality of tuning of the stochastic system to the periodic input.

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 tun-

ing such as the spectral power amplification or signal-to-noise ratio, due to the

impact of small random oscillations near the equilibria, are not robust with respect

to dimension reduction. Comparing optimal tuning rates for the unreduced (dif-

fusion) model and the associated reduced (finite state Markov chain) model one

gets essentially different tuning scenarios. We therefore propose—in arbitrary fi-

nite space dimension—measures of quality of periodic tuning based uniquely on

the transition dynamics and show that these measures are robust. Via our central

space-time large deviations result they are able to explain stochastic resonance as

optimal tuning.

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. It does not aim at a comprehensive pre-

sentation of the many facets of stochastic resonance in various areas of sciences. In

particular it does not touch computational aspects relevant in particular in high

dimensions where analytical methods alone are too complex to be of practical use

any more.

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