arXiv subjects

Rafael R. Souza

Publications and source records attributed to Rafael R. Souza.

12 recordsLinked to original sources

arXiv2025

The Hutchinson-Barnsley theory for iterated function systems with general measures

In this work we present iterated function systems with general measures(IFSm) formed by a set of maps $τ_λ$ acting over a compact space $X$, for a compact space of indices, $Λ$. The Markov process $Z_k$ associated to the IFS iteration is defined using a general family of probabilities measures $q_x$ on $Λ$, where $x \in X$: $Z_{k+1}$ is given by $τ_λ(Z_k)$, with $λ$ randomly chosen according to $q_x$. We prove the existence of the topological attractor and the existence of the invariant attracting measure for the Markov Process. We also prove that the support of the invariant measure is given by the attractor and results on the stochastic stability of the invariant measures, with respect to changes in the family $q_x$.

math.DS↗

arXiv2024

Symbiotic stars in X-rays IV: XMM-Newton, Swift and TESS observations

White dwarf symbiotic binaries are detected in X-rays with luminosities in the range of 10$^{30}$ to 10$^{34}$ lumcgs. Their X-ray emission arises either from the accretion disk boundary layer, from a region where the winds from both components collide or from nuclear burning on the white dwarf surface. In our continuous effort to identify X-ray emitting symbiotic stars, we studied four systems using observations from the Neil Gehrels Swift Observatory and XMM-Newton satellites in X-rays and from TESS in the optical. The X-ray spectra were fit with absorbed optically thin thermal plasma models, either single- or multitemperature with kT $<$ 8 keV for all targets. Based on the characteristics of their X-ray spectra, we classified BD Cam as possible $β$-type, V1261 Ori and CD -27 8661 as $δ$-type, and confirmed NQ Gem as $β$/$δ$-type. The $δ$-type X-ray emission most likely arise in the boundary layer of the accretion disk, while in the case of BD Cam, its mostly-soft emission originates from shocks, possibly between the red giant and WD/disk winds. In general, we have found that the observed X-ray emission is powered by accretion at a low accretion rate of about 10$^{-11}$ M$_{\odot}$ yr$^{-1}$. The low ratio of X-ray to optical luminosities, however indicates that the accretion-disk boundary layer is mostly optically thick and tends to emit in the far or extreme UV. The detection of flickering in optical data provides evidence of the existence of an accretion disk.

astro-ph.SR↗

arXiv2017

Exclusion process with slow boundary

We study the hydrodynamic and the hydrostatic behavior of the Simple Symmetric Exclusion Process with \emph{slow boundary}. The term \emph{slow boundary} means that particles can be born or die at the boundary sites, at a rate proportional to $N^{-θ}$, where $θ> 0$ and $N$ is the scaling parameter. In the bulk, the particles exchange rate is equal to $1$. In the hydrostatic scenario, we obtain three different linear profiles, depending on the value of the parameter $θ$; in the hydrodynamic scenario, we obtain that the time evolution of the spatial density of particles, in the diffusive scaling, is given by the weak solution of the heat equation, with boundary conditions that depend on $ θ$. If $θ\in(0,1)$, we get Dirichlet boundary conditions, (which is the same behavior if $θ=0$, see \cite{f}); if $θ=1$, we get Robin boundary conditions; and, if $θ\in(1,\infty)$, we get Neumann boundary conditions.

math.PR↗

arXiv2016

Nonhomogeneous quantum Markov chains and a notion of ergodicity

Motivated by a model presented by S. Gudder, we study a quantum generalization of Markov chains and discuss the relation between these maps and open quantum random walks, a class of quantum channels described by S. Attal et al. We consider processes which are nonhomogeneous in time, i.e., at each time step, a possibly distinct evolution kernel. Inspired by a spectral technique described by L. Saloff-Coste and J. Zúñiga, we define a notion of ergodicity for nonhomogeneous quantum Markov chains and describe a criterion for ergodicity of such objects in terms of singular values. As a consequence we obtain a quantum version of the classical probability result concerning the behavior of the columns (or rows) of the iterates of a stochastic matrix induced by a finite, irreducible, aperiodic Markov chain. We are also able to relate the ergodic property presented here with the notions of weak and uniform ergodicity known in the literature of noncommutative $L^1$-spaces.

quant-ph↗

arXiv2016

Open quantum random walks: ergodicity, hitting times, gambler's ruin and potential theory

In this work we study certain aspects of Open Quantum Random Walks (OQRWs), a class of quantum channels described by S. Attal et al. \cite{attal}. As a first objective we consider processes which are nonhomogeneous in time, i.e., at each time step, a possibly distinct evolution kernel. Inspired by a spectral technique described by L. Saloff-Coste and J. Zúñiga \cite{saloff}, we define a notion of ergodicity for finite nonhomogeneous quantum Markov chains and describe a criterion for ergodicity of such objects in terms of singular values. As a second objective, and based on a quantum trajectory approach, we study a notion of hitting time for OQRWs and we see that many constructions are variations of well-known classical probability results, with the density matrix degree of freedom on each site giving rise to systems which are seen to be nonclassical. In this way we are able to examine open quantum versions of the gambler's ruin, birth-and-death chain and a basic theorem on potential theory.

math-ph↗

arXiv2015

Ergodic and Thermodynamic Games

Let $T:X\to X $ and $S:Y \to Y$ be continuous maps defined on compact sets. Let $$φ_i(μ,ν)=\int_{X \times Y} A_i(x,y) dμ(x) dν(y)\;\;{for} \;\; i=1,2,$$ where $μ$ is $T$-invariant and $ν$ is $S$-invariant, be pay-off functions for a game (in the usual sense of game theory) between players that have the set of invariant measures for $T$ (player 1) and $S$ (player 2) as possible strategies. Our goal here is to establish the notion of Nash equilibrium point for the game defined by this pay-offs and strategies. The main tools came from ergodic optimization (as we are optimizing over the set of invariant measures) and thermodynamic formalism (when we add to the integrals above the entropy of measures in order to define a second case to be explored). Both cases are ergodic versions of non-cooperative games. We show the existence of Nash equilibrium points with two independent arguments. One of the arguments works for the case with entropy, and uses only tools of thermodynamical formalism, while the other, that works in the case without entropy but can be adapted to deal with both cases, uses the Kakutani fixed point. We also present examples and briefly discuss uniqueness (or lack of uniqueness). In the end we present a different example where players are allowed to collaborate. This final example show connections between cooperative games and ergodic transport.

math.DS↗

arXiv2015

On a class of quantum channels, open random walks and recurrence

We study a particular class of trace-preserving completely positive maps, called PQ-channels, for which classical and quantum evolutions are isolated in a certain sense. By combining open quantum random walks with a notion of recurrence, we are able to describe criteria for recurrence of the walk related to this class of channels. Positive recurrence for open walks is also discussed in this context.

math-ph↗

arXiv2013

Entropy and Variational Principle for one-dimensional Lattice Systems with a general a-priori probability: positive and zero temperature

We generalize several results of the classical theory of Thermodynamic Formalism by considering a compact metric space $M$ as the state space. We analyze the shift acting on $M^\mathbb{N}$ and consider a general a-priori probability for defining the Transfer (Ruelle) operator. We study potentials $A$ which can depend on the infinite set of coordinates in $M^\mathbb{N}.$ We define entropy and by its very nature it is always a nonpositive number. The concepts of entropy and transfer operator are linked. If M is not a finite set there exist Gibbs states with arbitrary negative value of entropy. Invariant probabilities with support in a fixed point will have entropy equal to minus infinity. In the case $M=S^1$, and the a-priori measure is Lebesgue $dx$, the infinite product of $dx$ on $(S^1)^\mathbb{N}$ will have zero entropy. We analyze the Pressure problem for a Hölder potential $A$ and its relation with eigenfunctions and eigenprobabilities of the Ruelle operator. Among other things we analyze the case where temperature goes to zero and we show some selection results. Our general setting can be adapted in order to analyze the Thermodynamic Formalism for the Bernoulli space with countable infinite symbols. Moreover, the so called XY model also fits under our setting. In this last case M is the unitary circle $S^1$. We explore the differentiable structure of $(S^1)^\mathbb{N}$ by considering potentials which are of class $C^1$ and we show some properties of the corresponding main eigenfunctions.

math.DS↗

arXiv2012

Continuous time finite state mean field games

In this paper we consider symmetric games where a large number of players can be in any one of d states. We derive a limiting mean field model and characterize its main properties. This mean field limit is a system of coupled ordinary differential equations with initial-terminal data. For this mean field problem we prove a trend to equilibrium theorem, that is convergence, in an appropriate limit, to stationary solutions. Then we study the $N+1$-player problem, which the mean field model attempts to approximate. Our main result is the convergence as $N\to \infty$ of the mean field model and an estimate of the rate of convergence. We end the paper with some further examples for potential mean field games.

math.OC↗

arXiv2011

Mean field limit of a continuous time finite state game

Mean field games is a recent area of study introduced by Lions and Lasry in a series of seminal papers in 2006. Mean field games model situations of competition between large number of rational agents that play non-cooperative dynamic games under certain symmetry assumptions. They key step is to develop a mean field model, in a similar way that what is done in statistical physics in order to construct a mathematically tractable model. A main question that arises in the study of such mean field problems is the rigorous justification of the mean field models by a limiting procedure. In this paper we consider the mean field limit of two-state Markov decision problem as the number of players $N\to \infty$. First we establish the existence and uniqueness of a symmetric partial information Markov perfect equilibrium. Then we derive a mean field model and characterize its main properties. This mean field limit is a system of coupled ordinary differential equations with initial-terminal data. Our main result is the convergence as $N\to \infty$ of the $N$ player game to the mean field model and an estimate of the rate of convergence.

math.OC↗

arXiv2009

Discrete mean field games

In this paper we study a mean field model for discrete time, finite number of states, dynamic games. These models arise in situations that involve a very large number of agents moving from state to state according to certain optimality criteria. The mean field approach for optimal control and differential games (continuous state and time) was introduced by Lasry and Lions. Here we consider a discrete version of the problem. Our setting is the following: we assume that there is a very large number of identical agents which can be in a finite number of states. Because the number of agents is very large, we assume the mean field hypothesis, that is, that the only relevant information for the global evolution is the fraction $π^n_i$ of players in each state $i$ at time $n$. The agents look for minimizing a running cost, which depends on $π$, plus a terminal cost $V^N$. In contrast with optimal control, where usually only the terminal cost $V^N$ is necessary to solve the problem, in mean-field games both the initial distribution of agents $π^0$ and the terminal cost $V^N$ are necessary to determine the solutions, that is, the distribution of players $π^n$ and value function $V^n$, for $0\leq n\leq N$. Because both initial and terminal data needs to be specified, we call this problem the initial-terminal value problem. Existence of solutions is non-trivial. Nevertheless, following the ideas of Lasry and Lions, we were able to establish existence and uniqueness, both for the stationary and for the initial-terminal value problems. In the last part of the paper we prove the main result of the paper, namely the exponential convergence to a stationary solution of $(π^0, V^0)$, as $N\to \infty$, for the initial-terminal value problem with (fixed) data $π^{-N}$ and $V^N$.

math.OC↗

arXiv2009

Negative Entropy, Zero temperature and stationary Markov Chains on the interval

We analyze some properties of maximizing stationary Markov probabilities on the Bernoulli space $[0,1]^\mathbb{N}$, More precisely, we consider ergodic optimization for a continuous potential $A$, where $A: [0,1]^\mathbb{N}\to \mathbb{R}$ which depends only on the two first coordinates. We are interested in finding stationary Markov probabilities $μ_\infty$ on $ [0,1]^\mathbb{N}$ that maximize the value $ \int A d μ,$ among all stationary Markov probabilities $μ$ on $[0,1]^\mathbb{N}$. This problem correspond in Statistical Mechanics to the zero temperature case for the interaction described by the potential $A$. The main purpose of this paper is to show, under the hypothesis of uniqueness of the maximizing probability, a Large Deviation Principle for a family of absolutely continuous Markov probabilities $μ_β$ which weakly converges to $μ_\infty$. The probabilities $μ_β$ are obtained via an information we get from a Perron operator and they satisfy a variational principle similar to the pressure. Under the hypothesis of $A$ being $C^2$ and the twist condition, that is, $\frac{\partial^2 A}{\partial_x \partial_y} (x,y) \neq 0$, for all $(x,y) \in [0,1]^2$, we show the graph property.

math.DS↗