INTRODUCTION xv
and direct approach is Ciesielski’s isomorphism of normed spaces of continuous
functions with sequence spaces via Fourier representation. The proof of the LDP
for Brownian motion using these arguments is given after recalling general notions
and basic concepts about large deviations, especially addressing their construction
from exponential decay rates of probabilities of basis sets of topologies, and their
transport between different topological spaces via continuous mappings (contrac-
tion principle). Since we only consider diffusion processes with additive noise for
which Itˆ o’s map is continuous, an appeal to the contraction principle provides the
LDP for the homogeneous diffusion processes we study. Finally, we follow Dembo
and Zeitouni [25] to derive the exit time laws due to Freidlin and Wentzell [40] for
time homogeneous diffusions from domains of attraction of underlying dynamical
systems in the small noise limit.
Chapter 3 deals with an approach to stochastic resonance for diffusions with
weakly time periodic drift and additive noise in the spirit of the associated Mar-
kovian semigroups and their spectral theory. This approach, presented in space
dimension 1, is clearly motivated by the physical notions of periodic tuning, in
particular the spectral power amplification coefficient. It describes the average
spectral component of the diffusion trajectories corresponding to the frequency of
the periodic input signal given by the drift term. We first give a rigorous account
of Freidlin’s quasi-deterministic limiting motion for potential double well diffusions
of this type. We then follow the paradigm of the physics literature, in particular
NcNamara and Wiesenfeld [74], and introduce the effective dynamics of our weakly
periodically forced double well diffusions given by reduced continuous time Markov
chains jumping between their two meta-stable equilibria. In this setting, different
notions of periodic tuning can easily be investigated. We not only consider the
physicists’ favorites, spectral power amplification and signal-to-noise ratio, but also
other reasonable concepts in which the energy carried by the Markov chain trajecto-
ries or the entropy of their invariant measures are used. Turning to diffusions with
weakly time periodic double well potentials and additive noise again, we then de-
velop an asymptotic analysis of their spectral power amplification coefficient based
on the spectral theory of their infinitesimal generators. It is based on the crucial
observation that in the case of double well potentials its spectrum has a gap be-
tween the second and third eigenvalue. Therefore we have to give the corresponding
eigenvalues and eigenfunctions a more detailed study, in particular with respect to
their asymptotic behavior in the small noise limit. Its results then enable us to give
a related small noise asymptotic expansion both of the densities of the associated
invariant measures as for the spectral power amplification coefficients. We finally
compare spectral power amplification coefficients of the Markov chains describing
the reduced dynamics and the associated diffusions, to find that in the small noise
limit they may be essentially different, caused by the small oscillations catastrophe
near the potential wells’ bottoms.
This motivates us in Chapter 4 to look for notions of periodic tuning for the so-
lution trajectories of diffusions in spaces of arbitrary finite dimension with weakly
periodic drifts and additive small noise which do not exhibit this robustness de-
fect. We aim at notions related to the maximal probabilities that the random exit
or transition times between different domains of attraction of the underlying dy-
namical systems happen in time windows parametrized by free energy parameters
on an exponential scale. For the two-state Markov chains describing the effective
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