LARGE DEVIATION PRINCIPLES FOR SOME STOCHASTIC DYNAMICAL SYSTEMS WITH ASYMPTOTICALLY VANISHING NOISE

We study three problems. First, we consider the large deviationbehavior of empirical measures of certain diffusion processes as, simultaneously,the time horizon becomes large and noise becomes vanishingly small. The lawof large numbers (LLN) of the empirical measure in this asymptotic regime isgiven by the unique equilibrium of the noiseless dynamics. Due to degeneracyof the noise in the limit, the methods of Donsker and Varadhan are not directly applicable and new ideasare needed. Second, we study a system of slow-fast diffusions where both theslow and the fast components have vanishing noise on their natural time scales.This time the LLN is governed by a degenerate averaging principle in whichlocal equilibria of the noiseless system obtained from the fast dynamics describethe asymptotic evolution of the slow component. We establish a large deviationprinciple that describes probabilities of divergence from this behavior. On theone hand our methods require stronger assumptions than the nondegeneratesettings, while on the other hand the rate functions take simple and explicitforms that have striking differences from their nondegenerate counterparts. Third, let $\Delta^o$ be a finite set and, for each probability measure $m$ on $\Delta^o$, let $G(m)$ be a transition kernel on $\Delta^o$. Consider the sequence $\{X_n\}$ of $\Delta^o$-valued random variables such that, given $X_0,\ldots,X_n$, the conditional distribution of $X_{n+1}$ is $G(L^{n+1})(X_n,\cdot)$, where $L^{n+1}=\frac{1}{n+1}\sum_{i=0}^{n}\delta_{X_i}$. Under conditions on $G$ we establish a large deviation principle for the sequence $\{L^n\}$. As one application of this result we obtain large deviation asymptotics for the Aldous-Flannery-Palacios (1988) approximation scheme for quasi-stationary distributions of finite state Markov chains. The conditions on $G$ cover other models as well, including certain models with edge or vertex reinforcement.