Institut Élie Cartan de Lorraine

About: Institut Élie Cartan de Lorraine is a facility organization based out in Vandœuvre-lès-Nancy, France. It is known for research contribution in the topics: Boundary value problem & Stochastic differential equation. The organization has 345 authors who have published 1084 publications receiving 15512 citations. The organization is also known as: Institut Élie-Cartan de Nancy.

Practical criteria for R-positive recurrence of unbounded semigroups

Abstract: The goal of this note is to show how recent results on the theory of quasi-stationary distributions allow to deduce effortlessly general criteria for the geometric convergence of normalized unbounded semigroups.

Minimization of the Eigenvalues of the Dirichlet-Laplacian with a Diameter Constraint

Abstract: In this paper we look for the domains minimizing the h-th eigenvalue of the Dirichlet-Laplacian λ h with a constraint on the diameter. Existence of an optimal domain is easily obtained, and is attained at a constant width body. In the case of a simple eigenvalue, we provide non standard (i.e., non local) optimality conditions. Then we address the question whether or not the disk is an optimal domain in the plane, and we give the precise list of the 17 eigenvalues for which the disk is a local minimum. We conclude by some numerical simulations showing the 20 first optimal domains in the plane.

Finite-time blowup and existence of global positive solutions of a semi-linear SPDE with fractional noise

Abstract: We consider stochastic equations of the prototype du(t, x) = delta u(t, x) + c*u(t, x) + u(t, x)^(1+ beta)) dt + k*u(t, x) dB^(H)_t on a smooth domain D in IR^d , with Dirichlet boundary condition, where beta > 0, c and k are constants and (B^(H)_ t , t in IR+) is a real-valued fractional Brownian motion with Hurst index H > 1/2. By means of an associated random partial differential equation lower and upper bounds for the blowup time are given. Sufficient conditions for blowup in finite time and for the existence of a global solution are deduced in terms of the parameters of the equation. For the case H = 1/2 (i.e. for Brownian motion) estimates for the probability of blowup in finite time are given in terms of the laws of exponential functionals of Brownian motion.

A general framework for waves in random media with long-range correlations

Abstract: We consider waves propagating in a randomly layered medium with long-range correlations. An example of such a medium is studied in \citeMS and leads, in particular, to an asymptotic travel time described in terms of a fractional Brownian motion. Here we study the asymptotic transmitted pulse under very general assumptions on the long-range correlations. In the framework that we introduce in this paper, we prove in particular that the asymptotic time-shift can be described in terms of non-Gaussian and/or multifractal processes.

Stochastic approximation on noncompact measure spaces and application to measure-valued Pólya processes

Abstract: Our main result is to prove almost-sure convergence of a stochastic-approximation algorithm defined on the space of measures on a non-compact space. Our motivation is to apply this result to measure-valued P\'olya processes (MVPPs, also known as infinitely-many P\'olya urns). Our main idea is to use Foster-Lyapunov type criteria in a novel way to generalize stochastic-approximation methods to measure-valued Markov processes with a non-compact underlying space, overcoming in a fairly general context one of the major difficulties of existing studies on this subject. From the MVPPs point of view, our result implies almost-sure convergence of a large class of MVPPs, this convergence was only obtained until now for specific examples, with only convergence in probability established for general classes. Furthermore, our approach allows us to extend the definition of MVPPs by adding "weights" to the different colors of the infinitely-many-color urn. We also exhibit a link between non-"balanced" MVPPs and quasi-stationary distributions of Markovian processes, which allows us to treat, for the first time in the literature, the non-balanced case. Finally, we show how our result can be applied to designing stochastic-approximation algorithms for the approximation of quasi-stationary distributions of discrete- and continuous-time Markov processes on non-compact spaces.