AVERAGING FOR A FULLY-COUPLED PIECEWISE DETERMINISTIC MARKOV PROCESS IN INFINITE DIMENSIONS

28 pagesIn this paper, we consider the generalized Hodgkin-Huxley model introduced by Austin in [1]. This model describes the propagation of an action potential along the axon of a neuron at the scale of ion channels. Mathematically, this model is a fully-coupled Piecewise Deterministic Markov Process (PDMP) in infinite dimensions. We introduce two time scales in this model in considering that some ion channels open and close at faster jump rates than others. We perform a slow-fast analysis of this model and prove that asymptotically this 'two-time-scales' model reduces to the so called averaged model which is still a PDMP in infinite dimensions for which we provide e ective evolution equations and jump rates

A convex programming approach for discrete-time Markov decision processes under the expected total reward criterion

In this work, we study discrete-time Markov decision processes (MDPs) under constraints with Borel state and action spaces and where all the performance functions have the same form of the expected total reward (ETR) criterion over the infinite time horizon. One of our objective is to propose a convex programming formulation for this type of MDPs. It will be shown that the values of the constrained control problem and the associated convex program coincide and that if there exists an optimal solution to the convex program then there exists a stationary randomized policy which is optimal for the MDP. It will be also shown that in the framework of constrained control problems, the supremum of the expected total rewards over the set of randomized policies is equal to the supremum of the expected total rewards over the set of stationary randomized policies. We consider standard hypotheses such as the so-called continuity-compactness conditions and a Slater-type condition. Our assumptions are quite weak to deal with cases that have not yet been addressed in the literature. An example is presented to illustrate our results with respect to those of the literature

Quina metodologia per despolhar l'enquèsta Bourciez ?

International audienceAn astonishing team of researchers, teachers and, above all, lovers of the Langue d'Oc with diverse and varied skills, has been carrying out a systematic analysis of the Bourciez survey for just over a year. In 1894, Edouard Bourciez, then a professor at the University of Bordeaux, asked teachers in the Bordeaux and Toulouse academies to translate a reworked version of the parable of the prodigal son into the idiom of the commune where they taught. The results of this investigation undoubtedly exceeded the professor's expectations, since more than 4400 parables were returned to him. These manuscripts are now kept at the University of Bordeaux and form a corpus of 17 volumes of about 1000 manuscript pages each. We are very grateful to the university library for allowing us to have them. For reasons that are not easy to explain, a systematic analysis of this survey has never been carried out, except for the Basque part corresponding to 150 municipalities, i.e. less than 4 percent of the corpus.We are proceeding with the analysis of this survey according to the following four points, not in series, but in parallel:- Computer transcription of the manuscripts ;- Creation of the database;- Data mining and statistical analysis of the database;- Formatting of the results in a format accessible to the greatest number of people

Averaging for slow-fast piecewise deterministic Markov processes with an attractive boundary

In this paper, we consider the problem of averaging for a class of piecewise deterministic Markov processes (PDMP) whose dynamic is constrained by the presence of a boundary. When reaching the boundary, the process is forced to jump away from it. We assume that this boundary is attractive for the process in question in the sense that its averaged flow is not tangent to it. Our averaging result relies strongly on the existence of densities for the process, allowing us to study the average number of crossings of a smooth hypersurface by an unconstrained PDMP and to deduce from this study averaging results for constrained PDMP

Self-similar behaviour of a non-local diffusion equation with time delay

We study the asymptotic behaviour of solutions of a class of linear non-local measure-valued differential equations with time delay. Our main result states that the solutions asymptotically exhibit a parabolic like behaviour in the large times, that is precisely expressed in term of heat kernel. Our proof relies on the study of a-self-similar-rescaled family of solutions. We first identify the asymptotic behaviour of the solutions by deriving a convergence result in the sense of the Young measures. Then we strengthen this convergence by deriving suitable fractional Sobolev compactness estimates. As a by-product, our main result allows to obtain asymptotic results for a class of piecewise constant stochastic processes with memory