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Large deviations principles of Non-Freidlin-Wentzell type.
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Joined: 08 Jul 2006
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PostPosted: Sat Jul 08, 2006 11:31 pm    Post subject: Large deviations principles of Non-Freidlin-Wentzell type. Reply with quote

We consider potential type dynamical systems in finite dimensions with two
meta-stable states. They are subject to two sources of perturbation: a
slow external periodic perturbation of period T and a small Gaussian
random perturbation of intensity s, and therefore mathematically described
as weakly time inhomogeneous diffusion processes. A system is in stochastic
resonance provided the small noisy perturbation is tuned in such a way that
its random trajectories follow the exterior periodic motion in an optimal
fashion, i.e. for some optimal intensity s(T). The physicists’ favorite
measures of quality of periodic tuning – and thus stochastic resonance –
such as spectral power amplification or signal-to-noise ratio have proven
to be defective. They are notrobust w.r.t. effective model reduction, i.e.
for the passage to a simplified finite state Markov chain model reducing
the dynamics to a pure jumping between the meta-stable states of the
original system. An entirely probabilistic notion of stochastic resonance
based on the transition dynamics between the domains of attraction of the
meta-stable states – and thus failing to su er from this robustness defect
– was proposed before in the context of one-dimensional diffusions. It is
investigated for higher dimensional systems here, by using extensions and
refinements of the Freidlin-Wentzell theory of large deviations for time
homogeneous di usions. Large deviation principles developed forweakly time
inhomogeneous di usions prove to be key tools for a treatment of the
problem of diffusion exit from a domain and thus for the approach of
stochastic resonance via transition probabilities between meta-stable
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