We present here a simplified version of recent results obtained with B. Helffer and M. Klein. They are concerned with the exponentally small eigenvalues of the Witten Laplacian on $0$-forms. We show how the Witten complex structure is better taken into account by working with singular values. This provides a convenient framework to derive accurate approximations of the first eigenvalues of ${\Delta }_{f,h}^{\left(0\right)}$ and solves efficiently the question of weakly resonant wells.

We consider the approximation of a mean field stochastic process by a large interacting particle system. We derive non-asymptotic large deviation bounds measuring the concentration of the empirical measure of the paths of the particles around the law of the process. The method is based on a coupling argument, strong integrability estimates on the paths in Hölder norm, and a general concentration result for the empirical measure of identically distributed independent paths.

We classify generators of quantum Markov semigroups on (h), with h finite-dimensional and with a faithful normal invariant state ρ satisfying the standard quantum detailed balance condition with an anti-unitary time reversal θ commuting with ρ, namely $tr\left({\rho }^{1/2}x{\rho }_t^{1/2}\left(y\right)\right)=tr({\rho }^{1/2}\theta y*\theta {\rho }_t^{1/2}(\theta x*\theta ))$ for all x,y ∈ and t ≥ 0. Our results also show that it is possible to find a standard form for the operators in the Lindblad representation of the generators extending the standard form of generators of quantum Markov semigroups satisfying the usual...

We review some recent results on quantum motion analysis, and in particular lower bounds for moments in quantum dynamics. The goal of the present exposition is to stress the role played by quantities we shall call Transport Integrals and by the so called generalized dimensions of the spectral measure in the analysis of quantum motion. We start with very simple derivations that illustrate how these quantities naturally enter the game. Then, gradually, we present successive improvements, up to most...

A one-dimensional quantum Euler-Poisson system for semiconductors for the electron density and the electrostatic potential in bounded intervals is considered. The existence and uniqueness of strong solutions with positive electron density is shown for quite general (possibly non-convex or non-monotone) pressure-density functions under a “subsonic” condition, i.e. assuming sufficiently small current densities. The proof is based on a reformulation of the dispersive third-order equation for the electron...

We considerably improve upon the recent result of [37] on the mixing time of Glauber dynamics for the 2D Ising model in a box of side $L$ at low temperature and with random boundary conditions whose distribution $P$ stochastically dominates the extremal plus phase. An important special case is when $P$ is concentrated on the homogeneous all-plus configuration, where the mixing time $T_{MIX}$ is conjectured to be polynomial in $L$. In [37] it was shown that for a large enough inverse-temperature $\beta$ and any $ϵ>0$ there...

Nous présentons une introduction à un nouveau champ de recherche, l’hypocoercitivité. Nous énonçons quelques résultats obtenus récemment avec différents co-auteurs (Lukas Neumann, Jean Dolbeault, Christian Schmeiser) dans le cas des équations cinétiques collisionnelles, en particulier pour les équations de type Boltzmann. Puis nous présentons quelques perspectives de recherche à plus long terme, dans le but de dégager une théorie unifiée de l’hypocoercitivité en théorie cinétique collisionnelle.

We consider a nearest-neighbor, one-dimensional random walk {Xn}n≥0 in a random i.i.d. environment, in the regime where the walk is transient with speed vP>0 and there exists an s∈(1, 2) such that the annealed law of n−1/s(Xn−nvP) converges to a stable law of parameter s. Under the quenched law (i.e., conditioned on the environment), we show that no limit laws are possible. In particular we show that there exist sequences {tk} and {tk'} depending on the environment only, such that a quenched...

For a sequence of i.i.d. random variables {ξx: x∈ℤ} bounded above and below by strictly positive finite constants, consider the nearest-neighbor one-dimensional simple exclusion process in which a particle at x (resp. x+1) jumps to x+1 (resp. x) at rate ξx. We examine a quenched non-equilibrium central limit theorem for the position of a tagged particle in the exclusion process with bond disorder {ξx: x∈ℤ}. We prove that the position of the tagged particle converges under diffusive scaling to a...