Invariant distributions and scaling limits for some diffusions in time-varying random environments

32 pagesInternational audienceWe consider a family of one-dimensional diffusions, in dynamical Wiener mediums, which are random perturbations of the Ornstein-Uhlenbeck diffusion process. We prove quenched and annealed convergences in distribution and under weighted total variation norms. We find two kind of stationary probability measures, which are either the standard normal distribution or a quasi-invariant measure, depending on the environment, and which is naturally connected to a random dynamical system. We apply these results to the study of a model of time-inhomogeneous Brox's diffusions, which generalizes the diffusion studied by Brox (1986) and those investigated by Gradinaru and Offret (2011). We point out two distinct diffusive behaviours and we give the speed of convergences in the quenched situations

Existence and asymptotic behaviour of some time-inhomogeneous diffusions

29 pagesInternational audienceLet us consider a solution of the time-inhomogeneous stochastic differential equation driven by a Brownian motion with drift coefficient $b(t,x)=\rho\,{\rm sgn}(x)|x|^\alpha/t^\beta$. This process can be viewed as a distorted Brownian motion in a potential, possibly singular, depending on time. After obtaining results on existence and uniqueness of solution, we study its asymptotic behaviour and made a precise description, in terms of parameters $\rho,\alpha$ and $\beta$, of the recurrence, transience and convergence. More precisely, asymptotic distributions, iterated logarithm type laws and rates of transience and explosion are proved for such processes

Invariant distributions and scaling limits for some diffusions in time-varying random environments

We consider a family of one-dimensional diffusions, in dynamical Wiener mediums, which are random perturbations of the Ornstein-Uhlenbeck diffusion process. We prove quenched and annealed convergences in distribution and under weigh-ted total variation norms. We find two kind of stationary probability measures, which are either the standard normal distribution or a quasi-invariant measure, depending on the environment, and which is naturally connected to a random dynamical system. We apply these results to the study of a model of time-inhomogeneous Brox's diffusions, which generalizes the diffusion studied by Brox (Ann Probab 14(4):1206-1218, 1986) and those investigated by Gradinaru and Offret (Ann Inst Henri Poincaré Probab Stat, 2011). We point out two distinct diffusive behaviours and we give the speed of convergences in the quenched situations

Comportement asymptotique d'une famille de diffusions inhomogènes en temps

National audienceSoit X la solution d'une équation différentielle stochastique dirigée par un mouvement Brownien B et ayant pour drift ..

A case of acute retinal pigment epithelitis: spectral domain optical coherence tomography time course and physiopathologic hypothesis

PURPOSE: To report the time course of retinal morphologic changes in a patient with acute retinal pigment epithelitis (ARPE) using spectral domain optical coherence tomography (SD-OCT). METHODS: A 30-year old man was referred for blurred vision of his right eye after five days that appeared suddenly 15 days after recovery from a flu-like syndrome. SD-OCT was performed immediately, followed by fluorescein and infracyanine angiography at eight days and then at three weeks. RESULTS: At presentation, a bubble of sub-macular deposit was observed on the right macula with central golden micronodules in a honeycomb pattern. SD-OCT showed an "anterior dislocation" of all the retinal layers up to the inner/outer segment (IS/OS) line and irregular deposits at the OS level together with thickening of the retinal pigment epithelial (RPE) layer. As visual acuity increased, eight days later, the OCT showed reduction of the sub-retinal deposits and an abnormal hyperflectivity of the sub-retinal and RPE layers was observed. The patient showed a positive serology for picornavirus. DISCUSSION: The acute SD-OCT sections of this patient with ARPE were compared with histological sections of a 35 day old Royal College of Surgeons rat. Similar findings could be observed, with preservation of the IS/OS line and accumulation of debris at the OS level, suggesting that ARPE symptoms could result from a transient phagocytic dysfunction of the RPE at the fovea, inducing reversible accumulation of undigested OS. Picornaviruses comprising enterovirus and coxsachievirus described as being associated with acute chorioretinitis. In this case, it was responsible for ARPE. CONCLUSION: We hypothesize that ARPE syndrome results from a transient dysfunction of RPE, which can occur as a post viral reaction

Morris Udall and his Favorite typewriter

Mo in front of his favorite typewrite

Dynamique de diffusions inhomogènes sous des conditions d'invariance d'échelle

We study the asymptotic behaviour of some stochastic processes whose dynamics depends not only on position, but also time, and such that the diffusion term and the potential satisfy some scaling properties. We point out a general phase transition phenomenon, entirely determined by the self-similar parameters. The main idea is to consider an appropriate scaling transformation, taking full advantage of the scaling properties. In the first part, we investigate a family of one-dimensional diffusion processes, driven by a Brownian motion, whose drift is polynomial in time and space. These diffusions are continuous counterparts of the random walks studied by Menshikov and Volkov (2008) and related to theFriedman's urn model. We give, in terms of all scaling parameters, the iterated logarithm type laws, the scaling limits and the explosion times of these processes.The second part dealt with a family of diffusion processes in random environment, directed by a one dimensional Brownian motion, whose potential is Brownian in space and polynomialin time. This situation is a generalization of the time-homogeneous Brox's diffusion (86) studied in an extensive body of the literature. We obtain in the critical case a quasi-invariant and quasi stationary random measure for the time-inhomogeneous semi-group, deduced from the study of a underlying random dynamical system.Nous étudions le comportement en temps long de certains processus stochastiques dont la dynamique dépend non seulement de la position, mais aussi du temps, et dont le terme de diffusion et le potentiel satisfont des conditions d'invariance d'échelle. Nous mettons en lumière un phénomène de transition de phase générale, entièrement déterminé par les différents indices d'auto-similarité en jeu. La principale idée mise en exergue est de considérer une transformation d'échelle adéquate, tirant pleinement parti des nombreuses invariances de notre problème.Dans une première partie, nous étudions une famille de processus de diffusion unidimensionnels, dirigés par un mouvement brownien, dont la dérive est polynomiale en temps et en espace. Ces diffusions généralisent les marches aléatoires, en lien avec le modèle d'urne de Friedman, étudiées par Menshikov et Volkov (2008). Nous donnons, de manière exhaustive, les lois du type logarithme itéré, les limites d'échelle ainsi que les temps de survie de ces processus. La seconde partie est, quant à elle, consacrée à l'étude d'une famille de processus de diffusion en environnement aléatoire, dirigés par un mouvement brownien unidimensionnel, dont le potentiel est brownien en espace et polynomial en temps. Ces diffusions sont une extensiondu modèle amplement étudié de Brox (86) et, en un sens randomisé, du modèle précédent. La différence notable avec le modèle déterministe est que nous obtenons, dans le cas critique, une mesure aléatoire quasi-invariante et quasi-stationnaire pour le semi-groupe, déduite de l'étude d'un système dynamique aléatoire sous-jacent

Spotlight on Antimicrobial Metabolites from the Marine Bacteria Pseudoalteromonas: Chemodiversity and Ecological Significance

This review is dedicated to the antimicrobial metabolite-producing Pseudoalteromonas strains. The genus Pseudoalteromonas hosts 41 species, among which 16 are antimicrobial metabolite producers. To date, a total of 69 antimicrobial compounds belonging to 18 different families have been documented. They are classified into alkaloids, polyketides, and peptides. Finally as Pseudoalteromonas strains are frequently associated with macroorganisms, we can discuss the ecological significance of antimicrobial Pseudoalteromonas as part of the resident microbiota