Large Deviations for SDEs

Large Deviations concern about giving sharp logarithmic asymptotics as $\varepsilon \to 0$ for the probabilities $\mu^\varepsilon(A)$, where $\mu^\varepsilon$ is a family of probability measures on a metric space indexed by $\varepsilon>0$. We consider mostly the case where $\mu^\varepsilon$ is the law of a random process, solution to certain SDE, with noise intensity equal to $\varepsilon$. Starting from the classical Friedlin-Wentzell Theorem, that treats the case of drift $b$ bounded and Lipschitz continuous ,we find weaker sufficient conditions on $b$ which guarantee the validity of a Large Deviations Principle. Moreover, in the particular case $b(x) = x \|x\|^{\gamma-1}$, $\gamma \in (0,1)$, in addition to the previous one we establish a second Large Deviation Principle, strictly related to the Peano phenomenon