Poisson boundary of a relativistic diffusion in curved space-times: an example

We study in details the long-time asymptotic behavior of a relativistic diffusion taking values in the unitary tangent bundle of a curved Lorentzian manifold, namely a spatially flat and fast expanding Robertson-Walker space-time. We prove in particular that the Poisson boundary of the diffusion can be identified with the causal boundary of the underlying manifold.Comment: 16 pages, 2 figure

Asymptotic behavior of a relativistic diffusion in Robertson-Walker space-times

We determine the long-time asymptotic behavior of a relativistic diffusion taking values in the unitary tangent bundle of a Robertson-Walker space-time. We prove in particular that when approaching the explosion time of the diffusion, its projection on the base manifold almost surely converges to a random point of the causal boundary and we also describe the behavior of the tangent vector in the neighborhood of this limiting point.Comment: 42 pages, 6 figure

Universality of the mean number of real zeros of random trigonometric polynomials under a weak Cramer condition

We investigate the mean number of real zeros over an interval $[a,b]$ of a random trigonometric polynomial of the form $\sum_{k=1}^n a_k \cos(kt)+b_k \sin(kt)$ where the coefficients are i.i.d. random variables. Under mild assumptions on the law of the entries, we prove that this mean number is asymptotically equivalent to $\frac{n(b-a)}{\pi\sqrt{3}}$ as $n$ goes to infinity, as in the known case of standard Gaussian coefficients. Our principal requirement is a new Cramer type condition on the characteristic function of the entries which does not only hold for all continuous distributions but also for discrete ones in a generic sense. To our knowledge, this constitutes the first universality result concerning the mean number of zeros of random trigonometric polynomials. Besides, this is also the first time that one makes use of the celebrated Kac-Rice formula not only for continuous random variables as it was the case so far, but also for discrete ones. Beyond the proof of a non asymptotic version of Kac-Rice formula, our strategy consists in using suitable small ball estimates and Edgeworth expansions for the Kolmogorov metric under our new weak Cramer condition, which both constitute important byproducts of our approach

Trends to equilibrium for a class of relativistic diffusions

We address the question of the trends to equilibrium for a large class C of relativistic diffusions. We show the existence of a spectral gap using the Lyapounov method and deduce the exponential decay of the distance to equilibrium in L2-norm and in total variation. A similar result was obtained recently in arXiv:1009.5086 for a particular process of the class C.Comment: 10 page

Central Limit Theorem for a Class of Relativistic Diffusions

Two similar Minkowskian diffusions have been considered, on one hand by Barbachoux, Debbasch, Malik and Rivet ([BDR1], [BDR2], [BDR3], [DMR], [DR]), and on the other hand by Dunkel and H\"anggi ([DH1], [DH2]). We address here two questions, asked in [DR] and in ([DH1], [DH2]) respectively, about the asymptotic behaviour of such diffusions. More generally, we establish a central limit theorem for a class of Minkowskian diffusions, to which the two above ones belong. As a consequence, we correct a partially wrong guess in [DH1].Comment: 20 page

Kinetic Brownian motion on Riemannian manifolds

International audienceWe consider in this work a one parameter family of hypoelliptic diffusion processes on the unit tangent bundle T 1 M of a Riemannian manifold (M, g), collectively called kinetic Brownian motions, that are random perturbations of the geodesic flow, with a parameter σ quantifying the size of the noise. Projection on M of these processes provides random C 1 paths in M. We show, both qualitively and quantitatively, that the laws of these M-valued paths provide an interpolation between geodesic and Brownian motions. This qualitative description of kinetic Brownian motion as the parameter σ varies is complemented by a thourough study of its long time asymptotic behaviour on rotationally invariant manifolds, when σ is fixed, as we are able to give a complete description of its Poisson boundary in geometric terms

Diretrizes da World Federation of Societies of Biological Psychiatry (WFSBP) para tratamento biológico de transtornos depressivos unipolares, 1ª parte: tratamento agudo e de continuação do transtorno depressivo maior

These practice guidelines for the biological treatment of unipolar depressive disorders were developed by an international Task Force of the World Federation of Societies of Biological Psychiatry (WFSBP). The goal for developing these guidelines was to systematically review all available evidence pertaining to the treatment of unipolar depressive disorders, and to produce a series of practice recommendations that are clinically and scientifically meaningful based on the available evidence. These guidelines are intended for use by all physicians seeing and treating patients with these conditions. The data used for developing these guidelines have been extracted primarily from various national treatment guidelines and panels for depressive disorders, as well as from meta-analyses and reviews on the efficacy of antidepressant medications and other biological treatment interventions identified by a search of the MEDLINE database and Cochrane Library. The identified literature was evaluated with respect to the strength of evidence for its efficacy and was then categorized into four levels of evidence (A-D). This first part of the guidelines covers disease definition, classification, epidemiology and course of unipolar depressive disorders, as well as the management of the acute and continuation-phase treatment. These guidelines are primarily concerned with the biological treatment (including antidepressants, other psychopharmacological and hormonal medications, electroconvulsive therapy, light therapy, adjunctive and novel therapeutic strategies) of young adults and also, albeit to a lesser extent, children, adolescents and older adults.Estas diretrizes práticas para o tratamento biológico de transtornos depressivos unipolares foram desenvolvidas por uma Força-Tarefa internacional da Federação Mundial de Sociedades de Psiquiatria Biológica (WFSBP). O objetivo ao desenvolver tais diretrizes foi rever sistematicamente todas as evidências existentes referentes ao tratamento de transtornos depressivos unipolares e produzir uma série de recomendações práticas com significado clínico e científico, baseadas nas evidências existentes. Têm como objetivo seu uso por todos os médicos que atendam e tratem pacientes com essas afecções. Os dados usados para o desenvolvimento das diretrizes foram extraídos primariamente de várias diretrizes e painéis nacionais de tratamento para transtornos depressivos, bem como de metanálises e revisões sobre a eficácia dos antidepressivos e outras intervenções de tratamento biológico identificadas por uma busca no banco de dados MEDLINE e Cochrane Library. A literatura identificada foi avaliada quanto à força das evidências sobre sua eficácia e, então, categorizada em quatro níveis de evidências (A a D). Esta primeira parte das diretrizes abrange definição, classificação, epidemiologia e evolução dos transtornos depressivos unipolares, bem como tratamento das fases aguda e de manutenção. As diretrizes se referem primariamente ao tratamento biológico (incluindo antidepressivos, outros medicamentos psicofarmacológicos e hormonais, eletroconvulsoterapia, fototerapia, estratégias terapêuticas complementares e novas) de adultos jovens e também, embora em menor grau, de crianças, adolescentes e adultos idosos