Exit Problem and Stochastic Resonance for a Class of Random Perturbations

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## Abstract

The asymptotic exit problems for diffusion processes with small parameter were considered in the classic work of Freidlin and Wentzell. In 2000, a mathematical theory of stochastic resonance for systems with random perturbations was established by Freidlin in the frame of the large deviation theory.

This dissertation concerns exit problems and stochastic resonance
for a class of random perturbations approximating white noise. The
tools used in the proofs are the large deviation theory and the
Markov property of the processes. The first problem considered is
the exit problem and stochastic resonance for random perturbations
of random walks. It turns out that a specific random walk can be
chosen which approximates the large deviation asymptotics of the
Wiener process in the best way. Analogous results concerning exit
problems and stochastic resonance for this type of random
perturbations were obtained under appropriate assumptions and
compared with those of white noise type perturbation. The second
problem I consider is the exit problems for random perturbations
of a Gaussian process $\eta_{t}^{\mu,\varepsilon}$ which satisfies
the equation $ \mu \dot{\eta}*{t}^{\mu,\varepsilon}=-
\eta*{t}^{\mu,\varepsilon}+\sqrt{\varepsilon}\dot{W}*{t},
,\eta*{0}^{\mu,\varepsilon}=y, ,0<\mu<<1,,0<\varepsilon<<1 $.
One can check that $\int_{0}^{t} \eta_{s}^{\mu,\varepsilon}ds$
converges to $\sqrt{\varepsilon}W_{t}$ uniformly on $[0,T]$ in
probability as $\mu \downarrow 0$. Results concerning asymptotic
exit problems for this type of random perturbation were obtained
under appropriate assumptions. Since $\eta_{t}^{\mu,\varepsilon}$
is not a Markov process, this creates some difficulties for the
proof. A new Markov process was constructed and the Markov
property of the new process was used in the proof.