Aldéric Joulin

A new Poisson-type deviation inequality for Markov jump processes with positive Wasserstein curvature

The purpose of this paper is to extend the investigation of Poisson-type deviation inequalities started by Joulin (Bernoulli 13 (2007) 782--798) to the empirical mean of positively curved Markov jump processes. In particular, our main result generalizes the tail estimates given by Lezaud (Ann. Appl. Probab. 8 (1998) 849--867, ESAIM Probab. Statist. 5 (2001) 183--201). An application to birth--death processes completes this work.

Intertwining and commutation relations for birth-death processes

Given a birth-death process on N with semigroup (P_t) and a discrete gradient d_u depending on a positive weight u, we establish intertwining relations of the form d_u P_t = Q_t d_u, where (Q_t) is the Feynman-Kac semigroup with potential V_u of another birth-death process. We provide applications when V_u is positive and uniformly bounded from below, including Lipschitz contraction and Wasserstein curvature, various functional inequalities, and stochastic orderings. Our analysis is naturally connected to the previous works of Caputo-Dai Pra-Posta and of Chen on birth-death processes. The proofs are remarkably simple and rely on interpolation, commutation, and convexity.

Intertwining Relations for One-Dimensional Diffusions and Application to Functional Inequalities

Following the recent work Chafaï and Joulin (Bernoulli 19:1855–1879, 2013) fulfilled in the discrete case, we provide in this paper new intertwining relations for semigroups of one-dimensional diffusions. Various applications of these results are investigated, among them the famous variational formula of the spectral gap derived by Chen and Wang (Trans. Am. Math. Soc. 349:1239–1267, 1997) together with a new criterion ensuring that the logarithmic Sobolev inequality holds. We complete this work by revisiting some classical examples, for which new estimates on the optimal constants are derived.

A LOGARITHMIC SOBOLEV INEQUALITY FOR AN INTERACTING SPIN SYSTEM UNDER A GEOMETRIC REFERENCE MEASURE

Logarithmic Sobolev inequalities are an essential tool in the study of interacting particle systems, cf. e.g. 4, 5. In this note we show that the logarithmic Sobolev inequality proved on the configuration space NZ d under Poisson reference measures in 1 can be extended to geometric reference measures using the results of 2. As a corollary we obtain a deviation estimate for an interacting particle system.

A note on convex ordering for stable stochastic integrals

We establish a convex ordering between stochastic integrals driven by strictly α-stable processes with index α ∈ (1,2). Our approach is based on the forward–backward stochastic calculus for martingales together with a suitable decomposition of stable stochastic integrals.