dynamics of the diffusions with slow and weak time inhomogeneity this optimal
transition rate is readily calculated. This concept moreover has the advantage that
their related transition times, as well as the corresponding ones for diffusions with
a weak noise dependent time inhomogeneity, allow a treatment by methods of large
deviations in the small noise limit. We therefore start with a careful extension of
large deviations theory to diffusions with slow time inhomogeneity. The central
result for the subsequent analysis of their exit times is contained in a large devia-
tions principle, uniform with respect to the energy parameter. It allows us in the
sequel to derive upper and lower bounds for the asymptotic exponential exit rate
from domains of attraction for slowly time dependent diffusions. They combine
to the main large deviations result describing the exact asymptotic exponential
exit rates for slowly and weakly time inhomogeneous diffusions in the small noise
limit. This central result is tailor made for providing the optimal tuning rate re-
lated to maximal probability of transition during an exponential time window. We
finally compare the resulting stochastic resonance point to the ones obtained for
the Markov chains of the reduced dynamics, and conclude that they agree in the
small noise limit, thus establishing robustness.
In two appendices for easy reference in the text we collect some standard
results about Gronwall’s lemma and Laplace’s method for integrals with exponential
singularities of the integrand.
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