Large deviations for processes with discontinuous statistics

This paper is devoted to the problem of sample path large deviations for the Markov processes on R_+^N having a constant but different transition mechanism on each boundary set {x:x_i=0 for i\notin\Lambda, x_i>0 for i\in\Lambda}. The global sample path large deviation principle and an integral representation of the rate function are derived from local large deviation estimates. Our results complete the proof of Dupuis and Ellis of the sample path large deviation principle for Markov processes describing a general class of queueing networks.Comment: Published at http://dx.doi.org/10.1214/009117905000000189 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

The t-Martin boundary of reflected random walks on a half-space

The t-Martin boundary of a random walk on a half-space with reflected boundary conditions is identified. It is shown in particular that the t-Martin boundary of such a random walk is not stable in the following sense : for different values of t, the t-Martin compactifications are not homeomorphic to each other.Comment: 13 pages, 1 figur

Martin boundary of a killed random walk on a quadrant

A complete representation of the Martin boundary of killed random walks on the quadrant ${\mathbb{N}}^*\times{\mathbb{N}}^*$ is obtained. It is proved that the corresponding full Martin compactification of the quadrant ${\mathbb{N}}^*\times{\mathbb{N}}^*$ is homeomorphic to the closure of the set $\{w={z}/{(1+|z|)}:z\in{\mathbb{N}}^*\times{\mathbb{N}}^*\}$ in ${\mathbb{R}}^2$. The method is based on a ratio limit theorem for local processes and large deviation techniques.Comment: Published in at http://dx.doi.org/10.1214/09-AOP506 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

A Classification of Asymptotic Behaviors of Green Functions of Random Walks in the Quadrant

This paper investigates the asymptotic behavior of Green functions associated to partially homogeneous random walks in the quadrant $Z_+^2$. There are four possible distributions for the jumps of these processes, depending on the location of the starting point: in the interior, on the two positive axes of the boundary, and at the origin $(0,0)$. With mild conditions on the positive jumps of the random walk, which can be unbounded, a complete analysis of the asymptotic behavior of the Green function of the random walk killed at $(0,0)$ is achieved. The main result is that {\em eight} regions of the set of parameters determine completely the possible limiting behaviors of Green functions of these Markov chains. These regions are defined by a set of relations for several characteristics of the distributions of the jumps. In the transient case, a description of the Martin boundary is obtained and in the positive recurrent case, our results give the exact limiting behavior of the invariant distribution of a state whose norm goes to infinity along some asymptotic direction in the quadrant. These limit theorems extend results of the literature obtained, up to now, essentially for random walks whose jump sizes are either $0$ or $1$ on each coordinate. Our approach relies on a combination of several methods: probabilistic representations of solutions of analytical equations, Lyapounov functions, convex analysis, methods of homogeneous random walks, and complex analysis arguments