Some particular self-interacting diffusions: Ergodic behaviour and almost sure convergence

This paper deals with some self-interacting diffusions $(X_t,t\geq 0)$ living on $\mathbb{R}^d$. These diffusions are solutions to stochastic differential equations: $$\mathrm{d}X_t=\mathrm{d}B_t-g(t)\nabla V(X_t-\bar{\mu}_t)\,\mathrm{d}t,$$ where $\bar{\mu}_t$ is the empirical mean of the process $X$, $V$ is an asymptotically strictly convex potential and $g$ is a given function. We study the ergodic behaviour of $X$ and prove that it is strongly related to $g$. Actually, we show that $X$ is ergodic (in the limit quotient sense) if and only if $\bar{\mu}_t$ converges a.s. We also give some conditions (on $g$ and $V$) for the almost sure convergence of $X$.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ310 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

A counterexample to the Cantelli conjecture through the Skorokhod embedding problem

In this paper, we construct a counterexample to a question by Cantelli, asking whether there exists a nonconstant positive measurable function $\varphi$ such that for i.i.d. r.v. $X,Y$ of law $\mathcal{N}(0,1)$, the r.v. $X+\varphi(X)\cdot Y$ is also Gaussian. This construction is made by finding an unusual solution to the Skorokhod embedding problem (showing that the corresponding Brownian transport, contrary to the Root barrier, is not unique). To find it, we establish some sufficient conditions for the continuity of the Root barrier function.Comment: Published at http://dx.doi.org/10.1214/14-AOP932 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

On the stability and the exponential concentration of Extended Kalman-Bucy filters

The exponential stability and the concentration properties of a class of extended Kalman-Bucy filters are analyzed. New estimation concentration inequalities around partially observed signals are derived in terms of the stability properties of the filters. These non asymptotic exponential inequalities allow to design confidence interval type estimates in terms of the filter forgetting properties with respect to erroneous initial conditions. For uniformly stable signals, we also provide explicit non-asymptotic estimates for the exponential forgetting rate of the filters and the associated stochastic Riccati equations w.r.t. Frobenius norms. These non asymptotic exponential concentration and quantitative stability estimates seem to be the first results of this type for this class of nonlinear filters. Our techniques combine χ-square concentration inequalities and Laplace estimates with spectral and random matrices theory, and the non asymptotic stability theory of quadratic type stochastic processes

Self-interacting diffusions: long-time behaviour and exit-problem in the convex case

We study a class of time-inhomogeneous diffusion: the self-interacting one. We show a convergence result with a rate of convergence that does not depend on the diffusion coefficient. Finally, we establish a so-called Kramers' type law for the first exit-time of the process from domain of attractions when the landscapes are uniformly convex.Comment: arXiv admin note: substantial text overlap with arXiv:2201.1042

Génolevures: protein families and synteny among complete hemiascomycetous yeast proteomes and genomes

The Génolevures online database (http://cbi.labri.fr/Genolevures/ and http://genolevures.org/) provides exploratory tools and curated data sets relative to nine complete and seven partial genome sequences determined and manually annotated by the Génolevures Consortium, to facilitate comparative genomic studies of Hemiascomycete yeasts. The 2008 update to the Génolevures database provides four new genomes in complete (subtelomere to subtelomere) chromosome sequences, 50 000 protein-coding and tRNA genes, and in silico analyses for each gene element. A key element is a novel classification of conserved multi-species protein families and their use in detecting synteny, gene fusions and other aspects of genome remodeling in evolution. Our purpose is to release high-quality curated data from complete genomes, with a focus on the relations between genes, genomes and proteins