Improved exponential stability for near-integrable quasi-convex Hamiltonians

In this article, we improve previous results on exponential stability for analytic and Gevrey perturbations of quasi-convex integrable Hamiltonian systems. In particular, this provides a sharper upper bound on the speed of Arnold diffusion which we believe to be optimal

Remarks on step cocycles over rotations, centralizers and coboundaries

By using a cocycle generated by the step function $\varphi_{\beta, \gamma} = 1_{[0, \beta]} - 1_{[0, \beta]} (. + \gamma)$ over an irrational rotation $x \to x + \alpha \mod 1$, we present examples which illustrate different aspects of the general theory of cylinder maps. In particular, we construct non ergodic cocycles with ergodic compact quotients, cocycles generating an extension $T_{\alpha, \varphi}$ with a small centralizer. The constructions are related to diophantine properties of $\alpha, \beta, \gamma$

A tightness criterion for random fields, with application to the Ising model

We present a criterion for a family of random distributions to be tight in local H\"older and Besov spaces of possibly negative regularity. We then apply this criterion to the magnetization field of the two-dimensional Ising model at criticality, answering a question of Camia, Garban and Newman.Comment: 28 pages. EJP versio

Nodal solutions for the Choquard equation

We consider the general Choquard equations $$-\Delta u + u = (I_\alpha \ast |u|^p) |u|^{p - 2} u$$ where $I_\alpha$ is a Riesz potential. We construct minimal action odd solutions for $p \in (\frac{N + \alpha}{N}, \frac{N + \alpha}{N - 2})$ and minimal action nodal solutions for $p \in (2,\frac{N + \alpha}{N - 2})$. We introduce a new minimax principle for least action nodal solutions and we develop new concentration-compactness lemmas for sign-changing Palais--Smale sequences. The nonlinear Schr\"odinger equation, which is the nonlocal counterpart of the Choquard equation, does not have such solutions.Comment: 23 pages, revised version with additional details and symmetry properties of odd solution

Backward stochastic differential equation driven by a marked point process: An elementary approach with an application to optimal control

We address a class of backward stochastic differential equations on a bounded interval, where the driving noise is a marked, or multivariate, point process. Assuming that the jump times are totally inaccessible and a technical condition holds (see Assumption (A) below), we prove existence and uniqueness results under Lipschitz conditions on the coefficients. Some counter-examples show that our assumptions are indeed needed. We use a novel approach that allows reduction to a (finite or infinite) system of deterministic differential equations, thus avoiding the use of martingale representation theorems and allowing potential use of standard numerical methods. Finally, we apply the main results to solve an optimal control problem for a marked point process, formulated in a classical way.Comment: Published at http://dx.doi.org/10.1214/15-AAP1132 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org