**Mathematical Surveys and Monographs**

Volume: 194;
2014;
189 pp;
Hardcover

MSC: Primary 60;
Secondary 34; 37; 86

Print ISBN: 978-1-4704-1049-0

Product Code: SURV/194

List Price: $77.00

Individual Member Price: $61.60

**Electronic ISBN: 978-1-4704-1473-3
Product Code: SURV/194.E**

List Price: $77.00

Individual Member Price: $61.60

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#### Supplemental Materials

# Stochastic Resonance: A Mathematical Approach in the Small Noise Limit

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*Samuel Herrmann; Peter Imkeller; Ilya Pavlyukevich; Dierk Peithmann*

Stochastic resonance is a phenomenon arising
in a wide spectrum of areas in the sciences ranging from physics
through neuroscience to chemistry and biology.

This book presents a mathematical approach to stochastic resonance
which is based on a large deviations principle (LDP) for randomly
perturbed dynamical systems with a weak inhomogeneity given by an
exogenous periodicity of small frequency. Resonance, the optimal
tuning between period length and noise amplitude, is explained by
optimizing the LDP's rate function.

The authors show that not all physical measures of tuning quality
are robust with respect to dimension reduction. They propose measures
of tuning quality based on exponential transition rates explained by
large deviations techniques and show that these measures are
robust.

The book sheds some light on the shortcomings and strengths of
different concepts used in the theory and applications of stochastic
resonance without attempting to give a comprehensive overview of the
many facets of stochastic resonance in the various areas of
sciences. It is intended for researchers and graduate students in
mathematics and the sciences interested in stochastic dynamics who
wish to understand the conceptual background of stochastic resonance.

#### Table of Contents

# Table of Contents

## Stochastic Resonance: A Mathematical Approach in the Small Noise Limit

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Preface vii8 free
- Introduction ix10 free
- Heuristics of noise induced transitions 118 free
- Transitions for time homogeneous dynamical systems with small noise 2744
- Semiclassical theory of stochastic resonance in dimension 1 6986
- Large deviations and transitions between meta-stable states of dynamical systems with small noise and weak inhomogeneity 133150
- Supplementary tools 177194
- Laplace’s method 179196
- Bibliography 183200
- Index 189206 free
- Back Cover Back Cover1209

#### Readership

Graduate students and research mathematicians interested in large deviations and stochastic resonance.