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 optimising 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.
| Preface |
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vii | |
| Introduction |
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ix | |
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Chapter 1 Heuristics of noise induced transitions |
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1 | (26) |
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1.1 Energy balance models of climate dynamics |
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1 | (5) |
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1.2 Heuristics of our mathematical approach |
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6 | (8) |
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1.3 Markov chains for the effective dynamics and the physical paradigm of spectral power amplification |
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14 | (4) |
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1.4 Diffusions with continuously varying potentials |
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18 | (3) |
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1.5 Stochastic resonance in models from electronics to biology |
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21 | (6) |
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Chapter 2 Transitions for time homogeneous dynamical systems with small noise |
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27 | (42) |
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2.1 Brownian motion via Fourier series |
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28 | (9) |
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2.2 The large deviation principle |
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37 | (7) |
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2.3 Large deviations for Brownian motion |
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44 | (6) |
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2.4 The Freidlin-Wentzell theory |
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50 | (9) |
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2.5 Diffusion exit from a domain |
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59 | (10) |
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Chapter 3 Semiclassical theory of stochastic resonance in dimension 1 |
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69 | (64) |
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3.1 Freidlin's quasi-deterministic motion |
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69 | (9) |
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3.2 The reduced dynamics: stochastic resonance in two-state Markov chains |
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78 | (13) |
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3.3 Spectral analysis of the infinitesimal generator of small noise diffusion |
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91 | (23) |
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3.4 Semiclassical approach to stochastic resonance |
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114 | (19) |
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Chapter 4 Large deviations and transitions between meta-stable states of dynamical systems with small noise and weak inhomogeneity |
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133 | (44) |
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4.1 Large deviations for diffusions with weakly inhomogeneous coefficients |
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134 | (10) |
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4.2 A new measure of periodic tuning induced by Markov chains |
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144 | (10) |
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4.3 Exit and entrance times of domains of attraction |
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154 | (15) |
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4.4 The full dynamics: stochastic resonance in diffusions |
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169 | (8) |
| Appendix A Supplementary tools |
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177 | (2) |
| Appendix B Laplace's method |
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179 | (4) |
| Bibliography |
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183 | (6) |
| Index |
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189 | |
Samuel Herrmann, Université de Bourgogne, Dijon, France
Peter Imkeller, Humboldt-Universität zu Berlin, Germany
Ilya Pavlyukevich, Friedrich-Schiller-Universität Jena, Germany
Dierk Peithmann, Essen, Germany
