xvi INTRODUCTION

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.