Antonio Galves

Metastable behavior of stochastic dynamics: A pathwise approach

In this paper a new approach to metastability for stochastic dynamics is proposed. The basic idea is to study the statistics of each path, performing time averages along the evolution. Metastability would be characterized by the fact that the process of these time averages converges, under a suitable rescaling, to a measure valued Markov jump process. Here this convergence is shown for the Curie-Weiss mean field dynamics and also for a model with spatial structure: Harris contact process.

Repetition times for Gibbsian sources

In this paper we consider the class of stochastic stationary sources induced by one-dimensional Gibbs states, with Holder continuous potentials. We show that the time elapsed before the source repeats its first n symbols, when suitably renormalized, converges in law either to a log-normal distribution or to a finite mixture of exponential random variables. In the first case we also prove a large deviation result.

Hydrodynamic Limit for Interacting Neurons

This paper studies the hydrodynamic limit of a stochastic process describing the time evolution of a system with N neurons with mean-field interactions produced both by chemical and by electrical synapses. This system can be informally described as follows. Each neuron spikes randomly following a point process with rate depending on its membrane potential. At its spiking time, the membrane potential of the spiking neuron is reset to the value 0 and, simultaneously, the membrane potentials of the other neurons are increased by an amount of potential

Statistics of Close Visits to the Indifferent Fixed Point of an Interval Map

\frac{1}{N}

Fluctuations in Derrida's Random Energy and Generalized Random Energy Models

1N. This mimics the effect of chemical synapses. Additionally, the effect of electrical synapses is represented by a deterministic drift of all the membrane potentials towards the average value of the system. We show that, as the system size N diverges, the distribution of membrane potentials becomes deterministic and is described by a limit density which obeys a non linear PDE which is a conservation law of hyperbolic type.

Exponential Inequalities for Empirical Unbounded Context Trees

We consider a new class of non Markovian processes with a countable number of interacting components. At each time unit, each component can take two values, indicating if it has a spike or not at this precise moment. The system evolves as follows. For each component, the probability of having a spike at the next time unit depends on the entire time evolution of the system after the last spike time of the component. This class of systems extends in a non trivial way both the interacting particle systems, which are Markovian (Spitzer in Adv. Math. 5:246–290, 1970) and the stochastic chains with memory of variable length which have finite state space (Rissanen in IEEE Trans. Inf. Theory 29(5):656–664, 1983). These features make it suitable to describe the time evolution of biological neural systems. We construct a stationary version of the process by using a probabilistic tool which is a Kalikow-type decomposition either in random environment or in space-time. This construction implies uniqueness of the stationary process. Finally we consider the case where the interactions between components are given by a critical directed Erdös-Rényi-type random graph with a large but finite number of components. In this framework we obtain an explicit upper-bound for the correlation between successive inter-spike intervals which is compatible with previous empirical findings.

Speed of d-convergence for Markov approximations of chains with complete connections. A coupling approach☆

We study a dynamical system defined by a map of the interval [0, 1] which has 0 as an indifferent fixed point but is otherwise expanding. We prove that the sequence of successive entrance times in a small neighborhood [0,a] converges in law when suitably normalized to a homogeneous Poisson point process.

Chthamalus bisinuatus (Cirripedia) and Brachidontes solisianus (Bivalvia) spatial interactions: A stochastic model

We introduce a new criterion to select in a consistent way the probabilistic context tree generating a sample. The basic idea is to construct a totally ordered set of candidate trees. This set is composed by the “champion trees”, the ones that maximize the likelihood of the sample for each number of degrees of freedom. The smallest maximizer criterion selects the infimum of the subset of champion trees whose gain in likelihood is negligible. This study was motivated by the linguistic challenge of retrieving rhythmic patterns from written texts. Applied to a data set consisting of texts extracted from daily newspapers, our algorithm identifies different context trees for European Portuguese and Brazilian Portuguese. This is compatible with the long standing conjecture that European Portuguese and Brazilian Portuguese belong to different rhythmic classes. Moreover, these context trees have several interesting properties which are linguistically meaningful.