211 research outputs found
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
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