Searcharxiv⌕ Search

arXiv subjects

Irina Ignatiouk-Robert

Publications and source records attributed to Irina Ignatiouk-Robert.

14 recordsLinked to original sources
arXiv2023

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.

math.PR↗
arXiv2023

Asymptotics of the Green function of killed random walks in a cone of $\Z^d$

In this paper, we obtain the exact asymptotic behavior of Green functions of homogeneous random walks in $\Z^d$ killed at the first exit from and open cone of $\R^d$. Our approach combines methods of functional equations, integral representations of the Green function and Woess' approach for the case of homogeneous random walks in $\Z^d$.

math.PR↗
arXiv2022

Reflected random walks and unstable Martin boundary

We introduce a family of two-dimensional reflected random walks in the positive quadrant and study their Martin boundary. While the minimal boundary is systematically equal to a union of two points, the full Martin boundary exhibits an instability phenomenon, in the following sense: if some parameter associated to the model is rational (resp.\ non-rational), then the Martin boundary is countable, homeomorphic to $\mathbb Z\cup\{\pm\infty\}$ (resp.\ uncountable, homeomorphic to $\mathbb R\cup\{\pm\infty\}$). Such instability phenomena are very rare in the literature. Along the way of proving this result, we obtain several precise estimates for the Green functions of reflected random walks with escape probabilities along the boundary axes and an arbitrarily large number of inhomogeneity domains. Our methods mix probabilistic techniques and an analytic approach for random walks with large jumps in dimension two.

math.PR↗
arXiv2020

Martin boundary of a killed non-centered random walk in a general cone

We investigate Martin boundary for a non-centered random walk on ${\mathbb Z}^d$ killed up on the time $τ_\vartheta$ of the first exit from a convex cone with a vertex at $0$. The approach combines large deviation estimates, the ratio limite theorem and the ladder height process. The results are applied to identify the Martin boundary for a random walk killed upon the first exit from a convex cone having $C^1$ boundary.

math.PR↗
arXiv2019

Harmonic functions of random walks in a semigroup via ladder heights

We investigate harmonic functions and the convergence of the sequence of ratios $(P_x(τ_\vartheta {>} n)/P_e(τ_\vartheta {>} n))$ for a random walk on a countable group killed up on the time $τ_\vartheta$ of the first exit from some semi-group with an identity element $e$. Several results of classical renewal theory for one dimensional random walk killed at the first exit from the positive half-line are extended to a multi-dimensional setting. For this purpose, an analogue of the ladder height process and the corresponding renewal function $V$ are introduced. The results are applied to multidimensional random walks killed upon the times of first exit from a convex cone. Our approach combines large deviation estimates and an extension of Choquet-Deny theory.

math.PR↗
arXiv2012

Explicit Lyapunov functions and estimates of the essential spectral radius for Jackson networks

A family of explicit Lyapunov function for positive recurrent Markovian Jackson network is constructed. With this result we obtain explicit estimates of the tail distribution of the first time, when the process returns to large compact sets, and some explicit estimates of the essential spectral radius of the process. The essential spectral radius of the process provides the best geometric convergence rate to equilibrium that one can get by changing the transitions of the process in a finite set.

math.PR↗
arXiv2010

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.

math.PR↗
arXiv2009

Martin boundary of a killed random walk on $\Z_+^d$

The Martin compactification is investigated for a d-dimensional random walk which is killed when at least one of it's coordinates becomes zero or negative. The limits of the Martin kernel are represented in terms of the harmonic functions of the associated induced Markov chains. It is shown that any sequence of points x_n with lim_n |x_n| = \infty and lim_n x_n/|x_n|= q is fundamental in the Martin compactification if up to the multiplication by constants, the induced Markov chain corresponding to the direction q has a unique positive harmonic function. The full Martin compactification is obtained for Cartesian products of one-dimensional random walks. The methods involve a ratio limit theorem and a large deviation principle for sample paths of scaled processes leading to the logarithmic asymptotics of the Green function.

math.PR↗
arXiv2009

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.

math.PR↗
arXiv2009

Martin boundary of a reflected random walk on a half-space

The complete representation of the Martin compactification for reflected random walks on a half-space $\Z^d\times\N$ is obtained. It is shown that the full Martin compactification is in general not homeomorphic to the ``radial'' compactification obtained by Ney and Spitzer for the homogeneous random walks in $\Z^d$ : convergence of a sequence of points $z_n\in\Z^{d-1}\times\N$ to a point of on the Martin boundary does not imply convergence of the sequence $z_n/|z_n|$ on the unit sphere $S^d$. Our approach relies on the large deviation properties of the scaled processes and uses Pascal's method combined with the ratio limit theorem. The existence of non-radial limits is related to non-linear optimal large deviation trajectories.

math.PR↗
arXiv2007

Martin boundary of a killed random walk on a half-space

A complete representation of the Martin boundary of killed random walks on a half-space $\Z^{d-1}\times\N^*$ is obtained. In particular, it is proved that the corresponding Martin boundary is homemorphic to the half-sphere ${\cal S}^d_+ = \{z\in\R^{d-1}\times\R_+ : |z|=1\}$. The method is based on a combination of ratio limits theorems and large deviation techniques.

math.PR↗
arXiv2005

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\notinΛ, x_i>0 for i\inΛ}. 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.

math.PR↗
arXiv2004

On the spectrum of Markov semigroups via sample path large deviations

The essential spectral radius of a sub-Markovian process is defined as the infimum of the spectral radiuses of all local perturbations of the process. When the family of rescaled processes satisfies sample path large deviation principle, the spectral radius and the essential spectral radius are expressed in terms of the rate function. The paper is motivated by applications to reflected diffusions and jump Markov processes describing stochastic networks for which the sample path large deviation principle has been established and the rate function has been identified while essential spectral radius has not been calculated.

math.PR↗