Stochastic Resonance: A Mathematical Approach in the Small Noise Limit
About this Title
Samuel Herrmann, Université de Bourgogne, Dijon, France, Peter Imkeller, Humboldt-Universität zu Berlin, Berlin, Germany, Ilya Pavlyukevich, Friedrich-Schiller-Universität Jena, Jena, Germany and Dierk Peithmann, Essen, Germany
Publication: Mathematical Surveys and Monographs
Publication Year: 2014; Volume 194
ISBNs: 978-1-4704-1049-0 (print); 978-1-4704-1473-3 (online)
MathSciNet review: 3155413
MSC: Primary 60H10; Secondary 34F15, 37H10, 60F10, 60J60, 60J70, 65C50
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.
Table of Contents
- 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