Sur un ensemble fini, on s'intéresse aux minorations linéaires du trou spectral d'un noyau markovien réversible, en terme de la constante isopérimétrique associée. On montre que la constante optimale est l'inverse du cardinal de l'ensemble moins un, mais on verra aussi comment il est possible de l'améliorer dans certaines situations particulières (arbres pointés radiaux à nombre fini de générations). Une application des inégalités précédentes est de retrouver immédiatement le comportement...

The main topic of this talk is the speed estimation of stability/instability. The word “various” comes with no surprising since there are a lot of different types of stability/instability and each of them has its own natural distance to measure. However, the adjective “unified” is very much unexpected. The talk surveys our recent progress on the topic, made in the past five years or so.

Usually the problem of drift estimation for a diffusion process is considered under the hypothesis of ergodicity. It is less often considered under the hypothesis of null-recurrence, simply because there are fewer limit theorems and existing ones do not apply to the whole null-recurrent class. The aim of this paper is to provide some limit theorems for additive functionals and martingales of a general (ergodic or null) recurrent diffusion which would allow us to have a somewhat unified approach...

We study ergodic properties of stochastic dissipative systems with additive noise. We show that the system is uniformly exponentially ergodic provided the growth of nonlinearity at infinity is faster than linear. The abstract result is applied to the stochastic reaction diffusion equation in ${ℝ}^d$ with $d\le 3$.

Suppose that $𝒢$ is a finite, connected graph and $X$ is a lazy random walk on $𝒢$. The lamplighter chain $X^{\diamond }$ associated with $X$ is the random walk on the wreath product ${𝒢}^{\diamond }={𝐙}_2\wr 𝒢$, the graph whose vertices consist of pairs $(\underline{f},x)$ where $f$ is a labeling of the vertices of $𝒢$ by elements of ${𝐙}_2=\{0,1\}$ and $x$ is a vertex in $𝒢$. There is an edge between $(\underline{f},x)$ and $(\underline{g},y)$ in ${𝒢}^{\diamond }$ if and only if $x$ is adjacent to $y$ in $𝒢$ and $f_z=g_z$ for all $z\ne x,y$. In each step, $X^{\diamond }$ moves from a configuration $(\underline{f},x)$ by updating $x$ to $y$ using the transition rule of $X$ and then sampling both...

We improve and subsume the conditions of Johansson and Öberg and Berbee for uniqueness of a $g$-measure, i.e., a stationary distribution for chains with complete connections. In addition, we prove that these unique $g$-measures have Bernoulli natural extensions. We also conclude that we have convergence in the Wasserstein metric of the iterates of the adjoint transfer operator to the $g$-measure.

We prove that, up to scalar multiples, there exists only one local regular Dirichlet form on a generalized Sierpi´nski carpet that is invariant with respect to the local symmetries of the carpet. Consequently, for each such fractal the law of Brownian motion is uniquely determined and the Laplacian is well defined.

We construct a class of conformally invariant measures on sets (or paths) and we study the critical exponents called intersection exponents associated to these measures. We show that these exponents exist and that they correspond to intersection exponents between planar Brownian motions. More precisely, using the definitions and results of our paper [27], we show that any set defined under such a conformal invariant measure behaves exactly as a pack (containing maybe a non-integer number) of Brownian...

We consider the one-sided exit problem – also called one-sided barrier problem – for ($\alpha$-fractionally) integrated random walks and Lévy processes. Our main result is that there exists a positive, non-increasing function $\alpha \mapsto \theta \left(\alpha \right)$ such that the probability that any $\alpha$-fractionally integrated centered Lévy processes (or random walk) with some finite exponential moment stays below a fixed level until time $T$ behaves as $T^{-\theta \left(\alpha \right)+\mathrm{o}\left(1\right)}$ for large $T$. We also investigate when the fixed level can be replaced by a different barrier...

We present a lower and an upper bound for the second smallest eigenvalue of Laplacian matrices in terms of the averaged minimal cut of weighted graphs. This is used to obtain an upper bound for the real parts of the non-maximal eigenvalues of irreducible nonnegative matrices. The result can be applied to Markov chains.

The asymptotics of the probability that the self-intersection local time of a random walk on ${\mathbb{Z}}^d$ exceeds its expectation by a large amount is a fascinating subject because of its relation to some models from Statistical Mechanics, to large-deviation theory and variational analysis and because of the variety of the effects that can be observed. However, the proof of the upper bound is notoriously difficult and requires various sophisticated techniques. We survey some heuristics and some recently elaborated...