arXiv subjects

Christophe Gallesco

Publications and source records attributed to Christophe Gallesco.

18 recordsLinked to original sources

arXiv2026

Reduction and classification of higher-order Markov chains

We study the class structure of finite-alphabet Markov chains with arbitrary memory length. To capture the structural constraints induced by prohibited transitions, we introduce the skeleton of a higher-order transition kernel, defined as a reduced set of contexts encoding all essential zero-probability patterns. To each skeleton we associate a binary transition matrix. We show that the communicating class structure of this matrix completely determines the recurrent classes of the original higher-order Markov chain, along with their periods. As a consequence, simple criteria for essential irreducibility and periodicity follow directly from the skeleton, without constructing the full first-order representation on the enlarged state space. From a practical perspective, this approach can yield significant computational gains. An example illustrates how the skeleton may have substantially smaller order than the original chain.

math.ST↗

arXiv2025

Uniqueness of stationary compatible probability measures for chains of infinite order with forbidden transitions

In this paper, we consider chains of infinite order on countable state spaces with prohibited transitions. We give a set of sufficient conditions on the structure of the probability kernels of the chains to have at most one stationary probability measure compatible with the kernel. Our main result extends the uniqueness $\ell^2$ criterion from Johansson and \"Oberg (2003) which was obtained for strongly non-null chains. A particular attention is given to concrete examples, illustrating the main theorem and its corollaries, with comparison to results of the existing literature.

math.PR↗

arXiv2024

Convergence and Stability Analysis of the Extended Infinite Horizon Model Predictive Control

Model Predictive Control (MPC) is a popular technology to operate industrial systems. It refers to a class of control algorithms that use an explicit model of the system to obtain the control action by minimizing a cost function. At each time step, MPC solves an optimization problem that minimizes the future deviation of the outputs which are calculated from the model. The solution of the optimization problem is a sequence of control inputs, the first input is applied to the system, and the optimization process is repeated at subsequent time steps. In the context of MPC, convergence and stability are fundamental issues. A common approach to obtain MPC stability is by setting the prediction horizon as infinite. For stable open-loop systems, the infinite horizon can be reduced to a finite horizon MPC with a terminal weight computed through the solution of a Lyapunov equation. This paper presents a rigorous analysis of convergence and stability of the extended nominally stable MPC developed by Odloak [Odloak, D. Extended robust model predictive control, AIChE J. 50 (8) (2004) 1824-1836] and the stable MPC with zone control [Gonz\'alez, A.H., Odloak, D. A stable MPC with zone control, J. Proc. Cont. 19 (2009) 110-122]. The mathematical proofs consider that the system is represented by a general gain matrix $D_0$, i.e., not necessarily regular, and they are developed for any input horizon $m$. The proofs are based on elementary geometric and algebraic tools and we believe that they can be adapted to the derived MPC approaches, as well as future studies.

math.OC↗

arXiv2021

Mixing rates for potentials of non-summable variations

Mixing rates and decay of correlations for dynamics defined by potentials with summable variations are well understood, but little is known for non-summable variations. In this paper, we exhibit upper bounds for these quantities in the case of dynamics defined by potentials with square summable variations. We obtain these bounds as corollaries of a new block coupling inequality between pair of dynamics starting with different histories. As applications of our results, we prove a new weak invariance principle and a Chernoff-type inequality.

math.DS↗

arXiv2019

An improved decoupling inequality for random interlacements

In this paper we obtain a decoupling feature of the random interlacements process $\mathcal{I}^u \subset \mathbb{Z}^d$, at level $u$, $d\geq 3$. More precisely, we show that the trace of the random interlacements process on two disjoint finite sets, $\textsf{F}$ and its translated $\textsf{F}+x$, can be coupled with high probability of success, when $\|x\|$ is large, with the trace of a process of independent excursions, which we call the noodle soup process. As a consequence, we obtain an upper bound on the covariance between two $[0,1]$-valued functions depending on the configuration of the random interlacements on $\textsf{F}$ and $\textsf{F}+x$, respectively. This improves a previous bound obtained by Sznitman in [12].

math.PR↗

arXiv2019

On uniform closeness of local times of Markov chains and i.i.d. sequences

In this paper we consider the field of local times of a discrete-time Markov chain on a general state space, and obtain uniform (in time) upper bounds on the total variation distance between this field and the one of a sequence of $n$ i.i.d. random variables with law given by the invariant measure of that Markov chain. The proof of this result uses a refinement of the soft local time method of [11].

math.PR↗

arXiv2016

Constrained information transmission on Erdös-Rényi graphs

We model the transmission of information of a message on the Erdös-Rény random graph with parameters $(n,p)$ and limited resources. The vertices of the graph represent servers that may broadcast a message at random. Each server has a random emission capital that decreases by one at each emission. We examine two natural dynamics: in the first dynamics, an informed server performs its attempts, then checks at each of them if the corresponding edge is open or not; in the second dynamics the informed server knows a priori who are its neighbors, and it performs all its attempts on its actual neighbors in the graph. In each case, we obtain first and second order asymptotics (law of large numbers and central limit theorem), when $n\to \infty$ and $p$ is fixed, for the final proportion of informed servers.

math.PR↗

arXiv2015

Characterization of the stability of chains associated with $g$-measures

In this paper we introduce a notion of asymptotic stability of a probability kernel, which we call dynamic uniqueness. We say that a kernel exhibits dynamic uniqueness if all the stochastic chains starting from a fixed past coincide on the future tail $σ$-algebra. We prove that the dynamic uniqueness is generally stronger than the usual notion of uniqueness for $g$-measures. Our main result shows that dynamic uniqueness is equivalent to the weak-$\ell^2$ summability condition on the kernel. This generalizes and strengthens the Johansson-Öberg $\ell^2$ criterion for uniqueness of $g$-measures. Finally, among other things, we prove that the weak-$\ell^2$ criterion implies $β$-mixing of the unique $g$-measure compatible with a regular kernel improving several results in the literature.

math.PR↗

arXiv2014

Explicit estimates in the Bramson-Kalikow model

The aim of the present article is to explicitly compute parameters for which the Bramson-Kalikow model exhibits phase-transition. The main ingredient of the proof is a simple new criterion for non-uniqueness of $g$-measures. We show that the existence of multiple $g$-measures compatible with a function $g$ can be proved by estimating the $\bar{d}$-distances between some suitably chosen Markov chains. The method is optimal for the important class of binary regular attractive functions, which includes the Bramson-Kalikow model.

math.PR↗

arXiv2013

On large deviations for the cover time of two-dimensional torus

Let $\mathcal{T}_n$ be the cover time of two-dimensional discrete torus $\mathbb{Z}^2_n=\mathbb{Z}^2/n\mathbb{Z}^2$. We prove that $\mathbb{P}[\mathcal{T}_n\leq \frac{4}πγn^2\ln^2 n]=\exp(-n^{2(1-\sqrtγ)+o(1)})$ for $γ\in (0,1)$. One of the main methods used in the proofs is the decoupling of the walker's trace into independent excursions by means of soft local times.

math.PR↗

arXiv2013

Localization for a random walk in slowly decreasing random potential

We consider a continuous time random walk $X$ in random environment on $\Z^+$ such that its potential can be approximated by the function $V: \R^+\to \R$ given by $V(x)=\sig W(x) -\frac{b}{1-\alf}x^{1-\alf}$ where $\sig W$ a Brownian motion with diffusion coefficient $\sig>0$ and parameters $b$, $\alf$ are such that $b>0$ and $0<\alf<1/2$. We show that $¶$-a.s.\ (where $¶$ is the averaged law) $\lim_{t\to \infty} \frac{X_t}{(C^*(\ln\ln t)^{-1}\ln t)^{\frac{1}{\alf}}}=1$ with $C^*=\frac{2\alf b}{\sig^2(1-2\alf)}$. In fact, we prove that by showing that there is a trap located around $(C^*(\ln\ln t)^{-1}\ln t)^{\frac{1}{\alf}}$ (with corrections of smaller order) where the particle typically stays up to time $t$. This is in sharp contrast to what happens in the "pure" Sinai's regime, where the location of this trap is random on the scale $\ln^2 t$.

math.PR↗

arXiv2013

Random walks with unbounded jumps among random conductances II: Conditional quenched CLT

math.PR↗

arXiv2013

A conditional quenched CLT for random walks among random conductances on $\mathbb{Z}^d$

Consider a random walk among random conductances on $\mathbb{Z}^d$ with $d\geq 2$. We study the quenched limit law under the usual diffusive scaling of the random walk conditioned to have its first coordinate positive. We show that the conditional limit law is the product of a Brownian meander and a $(d-1)$-dimensional Brownian motion.

math.PR↗

arXiv2012

On the moments of the meeting time of independent random walks in random environment

We consider, in the continuous time version, $γ$ independent random walks on $\mathbb{Z_+}$ in random environment in the Sinai's regime. Let $T_\gam$ be the first meeting time of one pair of the $γ$ random walks starting at different positions. We first show that the tail of the quenched distribution of $T_γ$, after a suitable rescaling, converges in probability, to some functional of the Brownian motion. Then we compute the law of this functional. Eventually, we obtain results about the moments of this meeting time. Being $\Eo$ the quenched expectation, we show that, for almost all environments $ω$, $\Eo[T_γ^{c}]$ is finite for $c<γ(γ-1)/2$ and infinite for $c>γ(γ-1)/2$.

math.PR↗

arXiv2012

Conditional and uniform quenched CLTs for one-dimensional random walks among random conductances

We study a one-dimensional random walk among random conductances, with unbounded jumps. Assuming the ergodicity of the collection of conductances and a few other technical conditions (uniform ellipticity and polynomial bounds on the tails of the jumps) we prove a quenched \textit{conditional} invariance principle for the random walk, under the condition that it remains positive until time $n$. As a corollary of this result, we study the effect of conditioning the random walk to exceed level $n$ before returning to 0 as $n\to \infty$. One of the main tools for proving these conditional limit laws is the \textit{uniform} quenched functional Central Limit Theorem, that states that the convergence is uniform with respect to the starting point, provided that the starting point is chosen in a certain interval around the origin.

math.PR↗

arXiv2012

Random walks with unbounded jumps among random conductances I: Uniform quenched CLT

math.PR↗

arXiv2010

Spiders in random environment

A spider consists of several, say $N$, particles. Particles can jump independently according to a random walk if the movement does not violate some given restriction rules. If the movement violates a rule it is not carried out. We consider random walk in random environment (RWRE) on $\Z$ as underlying random walk. We suppose the environment $ω=(ω_x)_{x \in \Z}$ to be elliptic, with positive drift and nestling, so that there exists a unique positive constant $κ$ such that $\E[((1-ω_0)/ω_0)^κ]=1$. The restriction rules are kept very general; we only assume transitivity and irreducibility of the spider. The main result is that the speed of a spider is positive if $κ/N>1$ and null if $κ/N<1$. In particular, if $κ/N <1$ a spider has null speed but the speed of a (single) RWRE is positive.

math.PR↗

arXiv2010

A note on spider walks

Spider walks are systems of interacting particles. The particles move independently as long as their movement do not violate some given rules describing the relative position of the particles; moves that violate the rules are not realized. The goal of this paper is to study qualitative properties, as recurrence, transience, ergodicity, and positive rate of escape of these Markov processes.

math.PR↗