On weak and strong large deviation principles for the empirical measure of random walks.

Abstract

This work is divided into two chapters. In the 竡rst chapter, we provide an introduction to the theory of large deviations and prove a weak Large Deviation Principle (LDP) for the empirical measure of the random walk with certain rates. To achieve this, we use the Parabolic Anderson Model (PAM) and the Gärtner-Ellis Theorem. In the second chapter we show that the empirical measure of certain continuous time random walks satis竡es a strong large deviation principle with respect to a topology introduced in [21] by Mukherjee and Varadhan. This topology is natural in models which exhibit an invariance with respect to spatial translations. Our result applies in particular to the case of simple random walk and complements the results obtained in [21] in which the large deviation principle has been established for the empirical measure of Brownian motion.