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# Ph. D. Defense- Pavlos Zoubouloglou

## 27 Mar @ 1:00 pm - 3:00 pm

# Ph. D. Defense- Pavlos Zoubouloglou

**27 Mar @ 1:00 pm – 3:00 pm**

**Under the direction of:**Amarjit Budhiraja.

**Title:**Large Deviation Principles for Some Stochastic Dynamical Systems With Asymptotically Vanishing Noise

**Abstract:**We study three problems. First, we consider the large deviation

behavior of empirical measures of certain diffusion processes as, simultaneously,

the time horizon becomes large and noise becomes vanishingly small. The law

of large numbers (LLN) of the empirical measure in this asymptotic regime is

given by the unique equilibrium of the noiseless dynamics. Due to degeneracy

of the noise in the limit, the methods of Donsker and Varadhan are not directly applicable and new ideas are needed.

Second, we study a system of slow-fast diffusions where both the

slow and the fast components have vanishing noise on their natural time scales.

This time the LLN is governed by a degenerate averaging principle in which

local equilibria of the noiseless system obtained from the fast dynamics describe

the asymptotic evolution of the slow component. We establish a large deviation

principle that describes probabilities of divergence from this behavior. On the

one hand our methods require stronger assumptions than the nondegenerate

settings, while on the other hand the rate functions take simple and explicit

forms 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.

Meeting ID: 942 2373 9353

Passcode: 297172