For a pair potential Φ in Euclidean space ℝ^d satisfying some natural and sufficiently general conditions in the sense of Penrose [1] and Poghosyan and Ueltschi [2] we define by means of the so-called Ursell kernel a function r which is shown to be the correlation function of a unique process G, the limiting Gibbs process for (Φ, ρ) with empty boundary conditions. This process is exhibited as a Gibbs process in the sense of Dobrushin, Lanford and Ruelle for a class of pair potentials, which contains classical stable and hard-core potentials that are called Penrose potentials here. Particularly, a class of positive potentials is included. Finally, for some class of Penrose potentials we show that G is the unique Gibbs process for Φ. We use the classical method of Kirkwood-Salsburg equations. A decisive role is played by a generalization of Ruelle’s estimate for correlation functions.
We study the distributions of the overlap number and the overlapping time that a tagged customer experiences in a network of two infinite server queues in series. While allowing general service time distributions in both queues, we specify the results in some detail for the cases of exponential and deterministic service time distributions. We furthermore outline how one can obtain joint overlap number distributions in more general feedforward networks of infinite server queues.
We present a review of some published results concerning evolution of the notion of (conventional) Bose-Einstein condensation (BEG) predicted in Ideal Bose-Gas (IBG) in 1925. The first step forward was the notion of the conventional generalised BEG a` la van den Berg-Lewis-Pule discovered in the IBG and later in certain models with interactions in 1980s. The next development was a discovery of non-conventional condensation of bosons that occurs only due to interaction. Finally we discuss an example of (non-ideal) the Weakly Imperfect Bose-Gas (WIBG), where the conventional and non-conventional condensates may coexist due to interaction.
In this paper we review the contribution of Evgueny (Euge`ne for the French community) Pechersky to image processing. Euge`ne discovered image processing as he was visiting Telecom Paris in 1994. During his stay at Telecom Paris and later on, during some visits and collaboration at INRIA Sophia Antipolis, his work was focused on designing multi-body interaction models to overcome the limits of the well known Ising and Potts model for image processing applications. Notably, the derived models were applied to different applications both in remote sensing and biomedical images. In this paper we summarize Euge`ne's contribution to the so-called Bayesian approach in image processing.
We prove a dispersive decay in weighted norms for solutions to 1D Schrödinger and Klein-Gordon equations with a δ-function potential.
The method of correlation functions is a fundamental tool for solving problems in mathematical statistical physics. It is used to prove the existence of a Gibbs random field with a given potential and to establish its various properties. Due to this, the problem of extending the method of correlation functions to broader classes of physical systems is very relevant. In our previous papers, the method of correlation functions was generalized in two directions: first, by considering spin systems, and second, by applying the concept of a transition energy field. The present paper continues the study of generalized correlation functions. Namely, we prove the group property of correlation functions, which represents a certain type of decay of dependencies between the components of the system.
We establish a theorem for stopped non-negative almost supermartingales, analogous to the stopped martingale theorem. Our result builds upon the definition of a stopped process and turns the proof of Proposition 1 in [4] into an application of this theorem. However, for its demonstration, we used fundamental concepts and theorems from probability theory, avoiding the use of local martingale properties as was done in [4]. Furthermore, we present an extended version of Doob’s maximal inequality reformulated for non-negative almost supermartingales
This paper investigates the asymptotic of the number of distinct words in a finite Markov chain driven model. We analyse the normalized and centered processes associated with the occurrence of distinct words in the model. Each state of the Markov chain is associated with its own unique infinite dictionary. At each state of the Markov chain, words are selected from the dictionary according to an infinite urn scheme. The probabilities in each infinite urn scheme satisfy the condition of regular variation. We use a combination of asymptotic techniques and results for Gaussian processes and derive the covariance structure of the limiting processes. The influence of stationary probabilities of the Markov chain on the normalization and scaling of these processes is explored in detail. Our findings provide new insights into the interaction between word frequencies and the stationary distribution in systems with pairwise disjoint dictionaries. These results are applicable to a wide range of stochastic systems, offering a deeper understanding of their limiting behaviour.
We study inverse and direct spectral problems for a class of operators H = D + K acting in ℓ^2(ℤ^d), d ≥ 1. Here D is a multiplication operator, and K generalizes a lattice Laplacian and is given by a matrix (K_xy)_(x,y∈ℤ^d) with K_xx ≡ 0 and |K_x,y| ≤ ε e^(-c|x-y|^γ), where γ ∈ (0; 1), c > 0, 0 < ε ≪ 1. We streamline the existing KAM (Kolmogorov-Arnold-Moser) techniques and extend prior results, obtained for a simpler case γ = 1, to the case of a subexponential decay of the off-diagonal entries K_xy.
We introduce the following model for the evolution of a population. At every discrete time n >= 0 exactly one individual is introduced in the population and is assigned a death probability cn sampled from C, a fixed probability distribution. We think of cn as a genetic marker of this individual. At every time n >= 1 every individual in the population dies or not independently of each other with its corresponding death probability cn. We show that the population size goes to infinity if and only if E(1/C) = infinity. This is in sharp contrast with the model with constant c and with the model in random environment (same random cn for all individuals at time n). Both of these models are always positive recurrent. Thus, it is really the randomness of individual c's that makes the population survive! We also study the point process associated with our model. We show that the limit point process has an accumulation point near 0 for the c's. For certain C distributions, including the uniform, the limit process properly rescaled is also shown to converge to a non-homogeneous Poisson process.
The aim of this paper is to construct stochastic processes allowing to obtain probabilistic representations of classical, weak or viscosity solutions of the forward Cauchy problem for several types of systems of nonlinear PDEs arising as viscous conservation and balance laws in various applications. The required stochastic processes are constructed as solutions of corresponding stochastic differential equations (SDEs) both forward and backward in time. Due to non-linearity of PDE systems under consideration additional relations must be added to the SDEs in order to obtain closed systems that can be studied independently. These relations are proved to generate probabilistic representations of the required solutions of the Cauchy problem for the original nonlinear PDE systems. Probabilistic representations are used to develop new numerical algorithms for approximation of classical and viscosity solutions to nonlinear PDEs.
We study indexes of diversity of the abundance of species when their proportions are organized as an exchangeable random partition and we take a sample from them. Firstly, we prove a general result: the sequence of Bayesian estimators of any integrable function defined on countable partitions of the unit interval is an integrable martingale that converges a.s. and in L^1 to the function, when the sample size diverges to infinity. Hence, the Bayesian estimator fluc- tuates as an integrable martingale. For the Poisson-Dirichlet Process, we study the estimators of the entropy and the Gini indexes in more detail. A series of results are devoted to revealing that the behavior of the Bayesian estimators share a number of similarities with the plug-in estimators. These include the a.s. limit behavior, but we also consider behaviors expressing local relations between these estimators. This is the case for the one-step difference of the conditional plug-in entropy of the individuals given that their species is known. We prove that it can be rephrased for the Bayesian entropy estimator and this gives a one-step difference between processes that does not jump only when a new species is found. Similar behavior is established for the Gini index.
Let $ν_1,ν_2,\dots$ be a sequence of probabilities on the nonnegative integers, and $X=(X_1,X_2, \dots)$ be a sequence of independent random variables $X_i$ with law $ν_i$. For $λ>0$ denote $Z^λ_i:= \sum_x λ^xν_i(x)$ and $λ^{\max}:= \sup\{λ>0: Z^λ_i<\infty \text{ for all }i\}$, and assume $λ^{\max}>1$. For $λ<λ^{\max}$, define the tilted probability $ν_i^λ(x):= λ^xν_i(x)/Z^λ_i$, and let $X^λ$ be a sequence of independent variables $X^λ_i$ with law $ν^λ_i$, and denote $S^λ_n:= X^λ_1+\dots+X^λ_n$, with $S_n=S^1_n$. Choose $λ^*\in(1,λ^{\max})$ and denote $R^*_n:= E (S^{λ^*}_n)$. The Gibbs Conditioning Principle (GCP) holds if $P(X\in\cdot|S_n>R^*_n)$ converges weakly to the law of $X^{λ^*}$, as $n\to\infty$. We prove the GCP for log-concave $ν_i$'s, meaning $ν_i(x+1)\,ν_i(x-1) \le ( ν_i(x))^2$, subject to a technical condition that prevents condensation. The canonical measures are the distributions of the first $n$ variables, conditioned on their sum being $k$. Efron's theorem states that for log-concave $ν_i$'s, the canonical measures are stochastically ordered with respect to $k$. This, in turn, leads to the ordering of the conditioned tilted measures $P(X^λ\in\cdot|S^λ_n>R^*_n)$ in terms of $λ$. This ordering is a fundamental component of our proof.
This paper proves a large deviation principle (LDP) for the stationary distribution of queue lengths in a sub critical generalised Jackson network assuming the Cramer condition on the interarrival and service times. The deviation function is given by the quasipotential.
This article focuses on the analysis of Reflected Backward Doubly Stochastic Differential Equations in the presence of inhomogeneous Lévy process noise (abbreviated as RBDSDELs). The first coefficient exhibits non-deterministic Lipschitz properties, while the second coefficient is characterized by Lipschitz condition. Our objective is to establish both the existence and uniqueness of a solution using the principles of Snell envelope theory and the Banach fixed-point theorem.
We consider a three-state solid-on-solid (SOS) model in the presence of nonzero external field on a Cayley tree. We find all the translation-invariant p-adic quasi Gibbs measures (TIpQGMs) of the model. Using the boundedness of these measures, we establish the occurrence of a phase transition.
A large class of backward doubly stochastic differential equations whose coefficients are linearly perturbed is observed in this paper. Their solutions are compared in the L^p-sense (p ≥ 2) with the solutions of the appropriate unperturbed equations of the same type, under the most global non-Lipschitz condition for the coefficients of the equations. Also, an interval [t̅(η), T] ⊂ [0, T], on which the L^p difference between the solutions of perturbed and unperturbed equations is less than a given value η, is established.
In this paper, a critical Galton-Watson branching process with state-dependent immigration Z = {Zₙ: n ≥ 0} is considered, where immigration is allowed to enter iff the previous generation was empty. We obtain the convergence rate of the harmonic moments of Zₙ firstly, and then the rate of convergence to zero as n → ∞ of P(|Zₙ₊₁/Zₙ − 1| > ε | Zₙ > 0) (ε > 0) is studied.
We analyze mean-field equations obtained for models motivated by a large station-based car-sharing system in France called Autolib’. The main focus is on a version where users reserve a parking space when they take a car. In a first model, the reservation of parking spaces is effective for all users (see [bourdais2020mean]) and capacity constraints are ignored. The model is carried out in thermodynamical limit, that is when the number N of stations and the number of cars Mₙ tend to infinity, with U = lim_(N→∞) Mₙ/N. This limit is described by Kolmogorov’s equations of a two-dimensional time-inhomogeneous Markov process depicting the numbers of reservations and cars at a station. It satisfies a non-linear differential system. We prove analytically that this system has a unique solution, which converges, as t → ∞, to an equilibrium point exponentially fast. Moreover, this equilibrium point corresponds to the stationary distribution of a two-queue tandem (reservations, cars), which is here always ergodic. The intensity factor of each queue has an explicit form obtained from an intrinsic mass conservation relationship. Two related models with capacity constraints are briefly presented in the last section: the simplest one with no reservation leads to a one-dimensional problem; the second one corresponds to our first model with finite total capacity K. Keywords: Markov Process, Queueing Systems, Thermodynamical limit, Mean- eld, Car sharing
In this paper we establish a diffusion limit for a multivariate continuous time Markov chain whose components are indexed by vertices of a finite graph. The components take values in a common finite set of non-negative integers and evolve subject to a graph based log-linear interaction. We show that if the set of common values of the components expands to the set of all non-negative integers, then a time-scaled and normalised version of the Markov chain converges to a system of interacting Ornstein-Uhlenbeck processes reflected at the origin. This limit is akin to heavy traffic limits in queueing (and our model can be naturally interpreted as a queueing model). Our proof draws on developments in queueing theory and relies on martingale methods.
