arXiv subjects

Sara Brofferio

Publications and source records attributed to Sara Brofferio.

16 recordsLinked to original sources

arXiv2024

Recurrence of multidimensional affine recursions in the critical case

We prove, under different natural hypotheses, that the random multidimensional affine recursion $X_n=A_nX_{n-1}+B_n\in\mathbb{R}^d, n \geq 1$, is recurrent in the critical case. In particular we cover the cases where the matrices $A_n$ are similarities, invertible, rank 1 or with non negative coefficients. These results are a consequence of a criterion of recurrence for a large class of affine recursions on $\mathbb R^d$, based on some moment assumptions of the so-called ``reverse norm control random variable".

math.PR↗

arXiv2024

Unique ergodicity for random noninvertible maps on an interval

In this short note, we investigate non-invertible stochastic dynamical systems on the unit interval $[0, 1]$. We provide a handy condition for unique ergodicity for systems that are injective in mean. On the other hand, we give concrete examples where unique ergodicity fails.

math.DS↗

arXiv2021

Asymptotically linear iterated function systems on the real line

Given a sequence of i.i.d. random functions $Ψ_{n}:\mathbb{R}\to\mathbb{R}$, $n\in\mathbb{N}$, we consider the iterated function system and Markov chain which is recursively defined by $X_{0}^{x}:=x$ and $X_{n}^{x}:=Ψ_{n-1}(X_{n-1}^{x})$ for $x\in\mathbb{R}$ and $n\in\mathbb{N}$. Under the two basic assumptions that the $Ψ_{n}$ are a.s. continuous at any point in $\mathbb{R}$ and asymptotically linear at the "endpoints" $\pm\infty$, we study the tail behavior of the stationary laws of such Markov chains by means of Markov renewal theory. Our approach provides an extension of Goldie's implicit renewal theory and can also be viewed as an adaptation of Kesten's work on products of random matrices to one-dimensional function systems as described. Our results have applications in quite different areas of applied probability like queuing theory, econometrics, mathematical finance and population dynamics. Our results have applications in quite different areas of applied probability like queuing theory, econometrics, mathematical finance and population dynamics, e.g. ARCH models and random logistic transforms.

math.PR↗

arXiv2020

On uniqueness of invariant measures for random walks on HOMEO(R)

We consider random walks on the group of orientation-preserving homeomorphisms of the real line ${\mathbb R}$. In particular, the fundamental question of uniqueness of an invariant measure of the generated process is raised. This problem was already studied by Choquet and Deny (1960) in the context of random walks generated by translations of the line. Nowadays the answer is quite well understood in general settings of strongly contractive systems. Here we focus on broader class of systems satisfying the conditions: recurrence, contraction and unbounded action. We prove that under these conditions the random process possesses a unique invariant Radon measure on ${\mathbb R}$. Our work can be viewed as a subsequent paper of Babillot et al. (1997) and Deroin et al. (2013).

math.PR↗

arXiv2020

On the affine recursion on $\mathbb R_+^d$

We fix $d \geq 2$ and denote $\mathcal S$ the semi-group of $d \times d$ matrices with non negative entries. We consider a sequence $(A_n, B_n)_{n \geq 1} $ of i. i. d. random variables with values in $\mathcal S\times \mathbb R_+^d$ and study the asymptotic behavior of the Markov chain $(X_n)_{n \geq 0}$ on $ \mathbb R_+^d$ defined by: \[ \forall n \geq 0, \qquad X_{n+1}=A_{n+1}X_n+B_{n+1}, \] where $X_0$ is a fixed random variable. We assume that the Lyapunov exponent of the matrices $A_n$ equals $0$ and prove, under quite general hypotheses, that there exists a unique (infinite) Radon measure $λ$ on $(\mathbb R^+)^d$ which is invariant for the chain $(X_n)_{n \geq 0}$. The existence of $λ$ relies on a recent work by T.D.C. Pham about fluctuations of the norm of product of random matrices . Its unicity is a consequence of a general property, called "local contractivity", highlighted about 20 years ago by M. Babillot, Ph. Bougerol et L. Elie in the case of the one dimensional affine recursion .

math.PR↗

arXiv2015

On unbounded invariant measures of stochastic dynamical systems

We consider stochastic dynamical systems on ${\mathbb{R}}$, that is, random processes defined by $X_n^x=Ψ_n(X_{n-1}^x)$, $X_0^x=x$, where $Ψ_n$ are i.i.d. random continuous transformations of some unbounded closed subset of ${\mathbb{R}}$. We assume here that $Ψ_n$ behaves asymptotically like $A_nx$, for some random positive number $A_n$ [the main example is the affine stochastic recursion $Ψ_n(x)=A_nx+B_n$]. Our aim is to describe invariant Radon measures of the process $X_n^x$ in the critical case, when ${\mathbb{E}}\log A_1=0$. We prove that those measures behave at infinity like $\frac{dx}{x}$. We study also the problem of uniqueness of the invariant measure. We improve previous results known for the affine recursions and generalize them to a larger class of stochastic dynamical systems which include, for instance, reflected random walks, stochastic dynamical systems on the unit interval $[0,1]$, additive Markov processes and a variant of the Galton--Watson process.

math.PR↗

arXiv2015

A construction of the measurable Poisson boundary: from discrete to continuous groups

Let $Γ$ be a dense countable subgroup of a locally compact continuous group $G$. Take a probability measure $μ$ on $Γ$. There are two natural spaces of harmonic functions: the space of $μ$-harmonic functions on the countable group $Γ$ and the space of $μ$-harmonic functions seen as functions on $G$ defined a.s. with respect to its Haar measure $λ$. This leads to two natural Poisson boundaries: the $Γ$-Poisson boundary and the $G$-Poisson boundary. Since boundaries on the countable group are quite well understood, a natural question is to ask how $G$-boundary is related to the $Γ$-boundary. In this paper we present a theoretical setting to build the $G$-Poisson boundary from the $Γ$-boundary. We apply this technics to build the Poisson boundary of the closure of the Baumslag-Solitar group in the group of real matrices. In particular we show that, under moment condition and in the case that the action on $\mathbf{R}$ is not contracting, this boundary is the $p$-solenoid.

math.PR↗

arXiv2011

Brownian motion and Harmonic functions on Sol(p,q)

The Lie group Sol(p,q) is the semidirect product induced by the action of the real numbers R on the plane R^2 which is given by (x,y) --> (exp{p z} x, exp{-q z} y), where z is in R. Viewing Sol(p,q) as a 3-dimensional manifold, it carries a natural Riemannian metric and Laplace-Beltrami operator. We add a linear drift term in the z-variable to the latter, and study the associated Brownian motion with drift. We derive a central limit theorem and compute the rate of escape. Also, we introduce the natural geometric compactification of Sol(p,q) and explain how Brownian motion converges almost surely to the boundary in the resulting topology. We also study all positive harmonic functions for the Laplacian with drift, and determine explicitly all minimal harmonic functions. All this is carried out with a strong emphasis on understanding and using the geometric features of Sol(p,q), and in particular the fact that it can be described as the horocyclic product of two hyperbolic planes with curvatures -p^2 and -q^2, respectively.

math.PR↗

arXiv2009

Poisson boundary of $GL_d(\Q)$

We construct the Poisson boundary for a random walk supported by the general linear group on the rational numbers as the product of flag manifolds over the $p$-adic fields. To this purpose, we prove a law of large numbers using the Oseledets' multiplicative ergodic theorem.

math.PR↗

arXiv2008

On the invariant measure of the random difference equation $X_n=A_n X_{n-1}+ B_n$ in the critical case

We consider the autoregressive model on $\R^d$ defined by the following stochastic recursion $X_n = A_n X_{n-1}+B_n$, where $\{(B_n,A_n)\}$ are i.i.d. random variables valued in $\R^d\times \R^+$. The critical case, when $\E\big[\log A_1\big]=0$, was studied by Babillot, Bougeorol and Elie, who proved that there exists a unique invariant Radon measure $ν$ for the Markov chain $\{X_n \}$. In the present paper we prove that the weak limit of properly dilated measure $ν$ exists and defines a homogeneous measure on $\R^d\setminus \{0\}$.

math.PR↗

arXiv2005

Internal Diffusion Limited Aggregation on discrete groups having exponential growth

The Internal Diffusion Limited Aggregation has been introduced by Diaconis and Fulton in 1991. It is a growth model defined on an infinite set and associated to a Markov chain on this set. We focus here on sets which are finitely generated groups with exponential growth. We prove a shape theorem for the Internal DLA on such groups associated to symmetric random walks. For that purpose, we introduce a new distance associated to the Green function, which happens to have some interesting properties. In the case of homogeneous trees, we also get the right order for the fluctuations of that model around its limiting shape.

math.PR↗

arXiv2005

Positive harmonic functions for semi-isotropic random walks on trees, lamplighter groups, and DL-graphs

We determine all positive harmonic functions for a large class of "semi-isotropic" random walks on the lamplighter group, i.e., the wreath product of the cyclic group of order q with the infinite cyclic group. This is possible via the geometric realization of a Cayley graph of that group as the Diestel-Leader graph DL(q,q). More generally, DL(q,r) is the horocyclic product of two homogeneous trees with respective degrees $q+1$ and $r+1$, and our result applies to all DL-graphs. This is based on a careful study of the minimal harmonic functions for semi-isotropic walks on trees.

math.PR↗

arXiv2004

Green kernel estimates and the full Martin boundary for random walks on lamplighter groups and Diestel-Leader graphs

We determine the precise asymptotic behaviour (in space) of the Green kernel of simple random walk with drift on the Diestel-Leader graph $DL(q,r)$, where $q,r \ge 2$. The latter is the horocyclic product of two homogeneous trees with respective degrees $q+1$ and $r+1$. When $q=r$, it is the Cayley graph of the wreath product (lamplighter group) ${\mathbb Z}_q \wr {\mathbb Z}$ with respect to a natural set of generators. We describe the full Martin compactification of these random walks on $DL$-graphs and, in particular, lamplighter groups. This completes and provides a better approach to previous results of Woess, who has determined all minimal positive harmonic functions.

math.PR↗

arXiv2004

Poisson boundary for finitely generated groups of rational affinities

The group of affine transformations with rational coefficients, $aff(Q)$, acts naturally on the real line, but also on the $p$-adic fields. The aim of this note is to show that all these actions are necessary and sufficient to represent bounded $μ$-harmonic functions for a probability measure $μ$ on $aff(Q)$ that is supported by a finitely generated sub-group, that is to describe the Poisson boundary.

math.PR↗

arXiv2004

The Poisson boundary of random rational affinities

The group of affine transformations with rational coefficients acts naturally on the real line, but also on the $p$-adic fields. The aim of this note is to show that, for random walks whose laws have a finite first moment, all these actions are necessary and sufficient to describe the Poisson boundary, which is in fact the product of all the fields that contract in mean.

math.PR↗

arXiv2003

Renewal theory on the oriented tree

The affine group of a tree is the group of the isometries of a homogeneous tree that fix an end of its boundary. Consider a probability measure on this group and the associated random walk. The main goal of this paper is to determine the accumulation points of the potential kernel at the infinity. In particular we show that under suitable regularity hypotheses this kernel can be continuously extended to the tree boundary and we determine the limit measures.

math.PR↗