Quantitative estimates for the long time behavior of an ergodic variant of the telegraph process

Motivated by stability questions on piecewise deterministic Markov models of bacterial chemotaxis, we study the long time behavior of a variant of the classic telegraph process having a non-constant jump rate that induces a drift towards the origin. We compute its invariant law and show exponential ergodicity, obtaining a quantitative control of the total variation distance to equilibrium at each instant of time. These results rely on an exact description of the excursions of the process away from the origin and on the explicit construction of an original coalescent coupling for both velocity and position. Sharpness of the obtained convergence rate is discussed.Comment: Definitive version of former paper "Quantitative estimates for the long time behavior of a PDMP describing the movement of bacteria", now accepted in Advances in Applied Probability. Presentation changed. A diffusive scaling limit result is added. Sharpness of the long-time convergence rate is discussed. 20 pages, 3 figure

Estimates for the density of a nonlinear Landau process

The aim of this paper is to obtain estimates for the density of the law of a specific nonlinear diffusion process at any positive bounded time. This process is issued from kinetic theory and is called Landau process, by analogy with the associated deterministic Fokker-Planck-Landau equation. It is not Markovian, its coefficients are not bounded and the diffusion matrix is degenerate. Nevertheless, the specific form of the diffusion matrix and the nonlinearity imply the non-degeneracy of the Malliavin matrix and then the existence and smoothness of the density. In order to obtain a lower bound for the density, the known results do not apply. However, our approach follows the main idea consisting in discretizing the interval time and developing a recursive method. To this aim, we prove and use refined results on conditional Malliavin calculus. The lower bound implies the positivity of the solution of the Landau equation, and partially answers to an analytical conjecture. We also obtain an upper bound for the density, which again leads to an unusual estimate due to the bad behavior of the coefficients

Long time behavior of telegraph processes under convex potentials

We study the long-time behavior of variants of the telegraph process with position-dependent jump-rates, which result in a monotone gradient-like drift toward the origin. We compute their invariant laws and obtain, via probabilistic couplings arguments, some quantitative estimates of the total variation distance to equilibrium. Our techniques extend ideas previously developed for a simplified piecewise deterministic Markov model of bacterial chemotaxis.Comment: 26 pages, 3 figure

On the Depletion Problem for an Insurance Risk Process: New Non-ruin Quantities in Collective Risk Theory

The field of risk theory has traditionally focused on ruin-related quantities. In particular, the socalled Expected Discounted Penalty Function has been the object of a thorough study over the years. Although interesting in their own right, ruin related quantities do not seem to capture path-dependent properties of the reserve. In this article we aim at presenting the probabilistic properties of drawdowns and the speed at which an insurance reserve depletes as a consequence of the risk exposure of the company. These new quantities are not ruin related yet they capture important features of an insurance position and we believe it can lead to the design of a meaningful risk measures. Studying drawdowns and speed of depletion for L\'evy insurance risk processes represent a novel and challenging concept in insurance mathematics. In this paper, all these concepts are formally introduced in an insurance setting. Moreover, using recent results in fluctuation theory for L\'evy processes, we derive expressions for the distribution of several quantities related to the depletion problem. Of particular interest are the distribution of drawdowns and the Laplace transform for the speed of depletion. These expressions are given for some examples of L\'evy insurance risk processes for which they can be calculated, in particular for the classical Cramer-Lundberg model.Comment: 23 pages, 4 figure

Statistical modeling of the influence of a visual distractor on the following eye-fixations

International audienceWe examined the influence of a visual distractor appearing during a fixation on the following fixations during natural exploration. It is known that new objects, congruent or incongruent with the scene, appearing during a fixation are fixated more than chance [Brockmole, J. R., & Henderson, J. M. (2008). Prioritizing new objects for eye fixation in real-world scenes: Effects of object-scene consistency. Vis. Cog., 16(2-3), 375-390]. In this study, we replicated this result using a Gabor patch for the appearing object, called a distractor because it was artificial and non-related to scenes. Besides, we wanted to quantify its influence on the exploration. A statistical model of the fixation density function was designed to analyze how the exploration was disrupted from and after the onset of the distractor. The model was composed of a linear weighted combination of different maps modeling three independent factors influencing gaze positions. We wondered whether fixation locations observed were rather due to the distractor or the saliency of the scenes. As expected, at the beginning of the exploration, fixation locations were not randomly chosen but influenced by the saliency of the scene and the distractor. The distractor onset strongly influenced fixations and this influence decreased with time

Optimal Vaccination Policy to Prevent Endemicity: A Stochastic Model

We examine here the effects of recurrent vaccination and waning immunity on the establishment of an endemic equilibrium in a population. An individual-based model that incorporates memory effects for transmission rate during infection and subsequent immunity is introduced, considering stochasticity at the individual level. By letting the population size going to infinity, we derive a set of equations describing the large scale behavior of the epidemic. The analysis of the model's equilibria reveals a criterion for the existence of an endemic equilibrium, which depends on the rate of immunity loss and the distribution of time between booster doses. The outcome of a vaccination policy in this context is influenced by the efficiency of the vaccine in blocking transmissions and the distribution pattern of booster doses within the population. Strategies with evenly spaced booster shots at the individual level prove to be more effective in preventing disease spread compared to irregularly spaced boosters, as longer intervals without vaccination increase susceptibility and facilitate more efficient disease transmission. We provide an expression for the critical fraction of the population required to adhere to the vaccination policy in order to eradicate the disease, that resembles a well-known threshold for preventing an outbreak with an imperfect vaccine. We also investigate the consequences of unequal vaccine access in a population and prove that, under reasonable assumptions, fair vaccine allocation is the optimal strategy to prevent endemicity.Comment: 49 pages, 7 figure

How a distractor influences fixations during the exploration of natural scenes

The distractor effect is a well-established means of studying different aspects of fixation pro-gramming during the exploration of visual scenes. In this study, we present a task-irrelevant distractor to participants during the free exploration of natural scenes. We investigate the con-trol and programming of fixations by analyzing fixation durations and locations, and the link between the two. We also propose a simple mixture model evaluated using the Expectation-Maximization algorithm to test the distractor effect on fixation locations, including fixations which did not land on the distractor. The model allows us to quantify the influence of a visual distractor on fixation location relative to scene saliency for all fixations, at distractor onset and during all subsequent exploration. The distractor effect is not just limited to the current fixa-tion, it continues to influence fixations during subsequent exploration. An abrupt change in the stimulus not only increases the duration of the current fixation, it also influences the location of the fixation which occurs immediately afterwards and to some extent, in function of the length of the change, the duration and location of any subsequent fixations. Overall, results from the eye movement analysis and the statistical model suggest that fixation durations and locations are both controlled by direct and indirect mechanisms

AMELIS, étude des usages d’un calendrier électronique mural par des personnes âgées et leurs aidants

National audienceLa France et le Canada font l'expérience d'un vieillissement de la population dans le cadre duquel les technologies comme le calendrier électronique Amelis, pourraient représenter des instruments intéressants d' accompagnement des aînés et de leurs aidants. Les théories de l' activité médiatisée par des instruments et de l ' acceptation des technologies nous apprennent cependant que l ' appropriation d ' une technologie par des utilisateurs est un processus situé influencé par de multiples facteurs. L ' objectif de cette recherche est d ' étudier ce processus afin d ' analyser comment les usages du calendrier Amelis pourront se développer dans l ' activité quotidienne de personnes âgées françaises et québecoises. Une première phase de recherche fera évoluer le calendrier Amelis de sa version actuelle à une version co-­‐conçue par des personnes âgées et l ' équipe de recherche. Ensuite , cette technologie sera implantée au domicile de personnes âgées volontaires. La mise à l ' épreuve du calendrier avec le réel permettra ainsi d ' analyser le développement des usages en contexte