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Stochastic Resonance: A Mathematical Approach in the Small Noise Limit

Samuel Herrmann Université de Bourgogne, Dijon, France
Peter Imkeller Humboldt-Universität zu Berlin, Berlin, Germany
Ilya Pavlyukevich Friedrich-Schiller-Universität Jena, Jena, Germany
Dierk Peithmann , Essen, Germany
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Hardcover ISBN: 978-1-4704-1049-0
Product Code: SURV/194
List Price: $82.00 MAA Member Price:$73.80
AMS Member Price: $65.60 Electronic ISBN: 978-1-4704-1473-3 Product Code: SURV/194.E List Price:$77.00
MAA Member Price: $69.30 AMS Member Price:$61.60
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AMS Member Price: $98.40 Click above image for expanded view Stochastic Resonance: A Mathematical Approach in the Small Noise Limit Samuel Herrmann Université de Bourgogne, Dijon, France Peter Imkeller Humboldt-Universität zu Berlin, Berlin, Germany Ilya Pavlyukevich Friedrich-Schiller-Universität Jena, Jena, Germany Dierk Peithmann , Essen, Germany Available Formats:  Hardcover ISBN: 978-1-4704-1049-0 Product Code: SURV/194  List Price:$82.00 MAA Member Price: $73.80 AMS Member Price:$65.60
 Electronic ISBN: 978-1-4704-1473-3 Product Code: SURV/194.E
 List Price: $77.00 MAA Member Price:$69.30 AMS Member Price: $61.60 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$123.00 MAA Member Price: $110.70 AMS Member Price:$98.40
• Book Details

Mathematical Surveys and Monographs
Volume: 1942014; 189 pp
MSC: Primary 60; Secondary 34; 37; 86;

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.

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

• Chapters
• Chapter 1. Heuristics of noise induced transitions
• Chapter 2. Transitions for time homogeneous dynamical systems with small noise
• Chapter 3. Semiclassical theory of stochastic resonance in dimension 1
• Chapter 4. Large deviations and transitions between meta-stable states of dynamical systems with small noise and weak inhomogeneity
• Appendix A. Supplementary tools
• Appendix B. Laplace’s method

• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 1942014; 189 pp
MSC: Primary 60; Secondary 34; 37; 86;

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.

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

• Chapters
• Chapter 1. Heuristics of noise induced transitions
• Chapter 2. Transitions for time homogeneous dynamical systems with small noise
• Chapter 3. Semiclassical theory of stochastic resonance in dimension 1
• Chapter 4. Large deviations and transitions between meta-stable states of dynamical systems with small noise and weak inhomogeneity
• Appendix A. Supplementary tools
• Appendix B. Laplace’s method
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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