xiv INTRODUCTION transitions as functions of time with exponentially decaying spikes near the integer multiples of the forcing periods. It has the big advantage of being robust for model reduction, i.e. the passage from the diffusion to the two state Markov chain describing its reduced dynamics. The techniques needed to prove this main result feature non-trivial extensions and refinements of the fundamental large deviations theory for time homogeneous diffusions by Freidlin–Wentzell [40]. We prove a large deviations principle for the inhomogeneous diffusion (0.1) and further strengthen this result to get uniformity in system parameters. Similarly to the time homogeneous case, where large deviations theory is applied to the problem of diffusion exit culminating in a mathematically rigorous proof of the Kramers–Eyring law, we study the problem of diffusion exit from a domain which is carefully chosen in order to allow for a detailed analysis of transition times. The main idea behind our analysis is that the natural time scale is so large that re-scaling in these units essentially leads to an asymptotic freezing of the time inhomogeneity, which has to be carefully controlled, to hook up to the theory of large deviations of time homogeneous diffusions. The material in the book is organized as follows. In Chapter 1 we give a de- tailed treatment of the heuristics behind our mathematical approach, mostly in space dimension 1. We start by giving a fairly thorough account of the paradigm of glacial cycles which was the historical root of physical models exhibiting sto- chastic resonance. It gives rise to the model equation of a weakly periodically forced dynamical system with noise that can be interpreted as the motion of an overdamped physical particle in a weakly periodically forced potential landscape subject to noise. The heuristics of exit and transition behavior between domains of attraction (potential wells) of such systems based on the classical large deviations theory is explained in two steps: first for time independent potential landscapes, then for potentials switching discontinuously between two anti-symmetric states every half period. Freidlin’s quasi-deterministic motion is seen to not cover the concept of optimal periodic tuning between weak periodic input and randomly am- plified output. They determine stochastic resonance through measures of quality of periodic tuning such as the spectral power amplification or the signal-to-noise ratio. The latter concepts are studied first for finite state Markov chains capturing the dynamics of the underlying diffusions reduced to the meta-stable states, and then for the diffusions with time continuous periodic potential functions. The robustness defect of the classical notions of resonance in passing from Markov chain to diffu- sion is pointed out. Then alternative notions of resonance are proposed which are based purely on the asymptotic behavior of transition times. Finally, examples of systems exhibiting stochastic resonance features from different areas of science are presented and briefly discussed. They document the ubiquity of the phenomenon of stochastic resonance. Our approach is based on concepts of large deviations. Therefore Chapter 2 is devoted to a self-contained treatment of the theory of large deviations for randomly perturbed dynamical systems in finite dimensions. Following a direct and elegant approach of Baldi and Roynette [3], we describe Brownian motion in its Schauder decomposition. It not only allows a direct approach to its regularity properties in terms of H¨ older norms on spaces of continuous functions. It also allows a derivation of Schilder’s large deviation principle (LDP) for Brownian motion from the elemen- tary LDP for one-dimensional Gaussian random variables. The key to this elegant

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