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.

Stable Determination of Time-Dependent Scalar Potential From Boundary Measurements in a Periodic Quantum Waveguide

Abstract: We prove logarithmic stability in the determination of the time-dependent scalar potential in a 1-periodic quantum cylindrical waveguide, from the boundary measurements of the solution to the dynamic Schrodinger equation.

Superrigidity in infinite dimension and finite rank via harmonic maps

Abstract: We prove geometric superrigidity for actions of cocompact lattices in semisimple Lie groups of higher rank on infinite dimensional Riemannian manifolds of nonpositive curvature and finite telescopic dimension

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

Abstract: This paper deals with some self-interacting diffusions $(X_t, t \geq 0)$ living on $ℝ^d$. These diffusions are solutions to stochastic differential equations: $$\mathrm{d}X_t = \mathrm{d}B_t − g(t)∇V(X_t − \overline μ_t) \mathrm{d}t,$$ where $\overline μ_t$ 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 $\overline μ_t$ converges a.s. We also give some conditions (on $g$ and $V$) for the almost sure convergence of $X$.

The number of open paths in oriented percolation

Abstract: We study the number $N_n$ of open paths of length $n$ in supercritical oriented percolation on $\Zd \times \N$, with $d \ge 1$. We prove that on the percolation event $\{\inf N_n>0\}$, $N_n^{1/n}$ almost surely converges to a positive deterministic constant. We also study the existence of directional limits. The proof relies on the introduction of adapted sequences of regenerating times, on subadditive arguments and on the properties of the coupled zone in supercritical oriented percolation.

Transition times and stochastic resonance for multidimensional diffusions with time periodic drift: A large deviations approach

Abstract: We consider potential type dynamical systems in finite dimensions with two meta-stable states. They are subject to two sources of perturbation: a slow external periodic perturbation of period $T$ and a small Gaussian random perturbation of intensity $\epsilon$, and, therefore, are mathematically described as weakly time inhomogeneous diffusion processes. A system is in stochastic resonance, provided the small noisy perturbation is tuned in such a way that its random trajectories follow the exterior periodic motion in an optimal fashion, that is, for some optimal intensity $\epsilon (T)$. The physicists' favorite, measures of quality of periodic tuning--and thus stochastic resonance--such as spectral power amplification or signal-to-noise ratio, have proven to be defective. They are not robust w.r.t. effective model reduction, that is, for the passage to a simplified finite state Markov chain model reducing the dynamics to a pure jumping between the meta-stable states of the original system. An entirely probabilistic notion of stochastic resonance based on the transition dynamics between the domains of attraction of the meta-stable states--and thus failing to suffer from this robustness defect--was proposed before in the context of one-dimensional diffusions. It is investigated for higher-dimensional systems here, by using extensions and refinements of the Freidlin--Wentzell theory of large deviations for time homogeneous diffusions. Large deviations principles developed for weakly time inhomogeneous diffusions prove to be key tools for a treatment of the problem of diffusion exit from a domain and thus for the approach of stochastic resonance via transition probabilities between meta-stable sets.