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 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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