Nonlinear PDEs with modulated dispersion II: Korteweg--de Vries equation

We continue the study of various nonlinear PDEs under the effect of a time--inhomogeneous and irregular modulation of the dispersive term. In this paper we consider the modulated versions of the 1d periodic or non-periodic Korteweg--de Vries (KdV) equation and of the modified KdV equation. For that we use a deterministic notion of "irregularity" for the modulation and obtain local and global results similar to those valid without modulation. In some cases the irregularity of the modulation improves the well-posedness theory of the equations. Our approach is based on estimates for the regularising effect of the modulated dispersion on the non-linear term using the theory of controlled paths and estimates stemming from Young's theory of integration.Comment: 37 page

Nonlinear PDEs with modulated dispersion I: Nonlinear Schr\"odinger equations

We start a study of various nonlinear PDEs under the effect of a modulation in time of the dispersive term. In particular in this paper we consider the modulated non-linear Schr\"odinger equation (NLS) in dimension 1 and 2 and the derivative NLS in dimension 1. We introduce a deterministic notion of "irregularity" for the modulation and obtain local and global results similar to those valid without modulation. In some situations, we show how the irregularity of the modulation improves the well--posedness theory of the equations. We develop two different approaches to the analysis of the effects of the modulation. A first approach is based on novel estimates for the regularising effect of the modulated dispersion on the non-linear term using the theory of controlled paths. A second approach is an extension of a Strichartz estimated first obtained by Debussche and Tsutsumi in the case of the Brownian modulation for the quintic NLS.Comment: 27 pages. Extensive reorganisation of the material and typos correcte

Skorohod and Stratonovich integration in the plane

This article gives an account on various aspects of stochastic calculus in the plane. Specifically, our aim is 3-fold: (i) Derive a pathwise change of variable formula for a path indexed by a square, satisfying some H\"older regularity conditions with a H\"older exponent greater than 1/3. (ii) Get some Skorohod change of variable formulas for a large class of Gaussian processes defined on the suqare. (iii) Compare the bidimensional integrals obtained with those two methods, computing explicit correction terms whenever possible. As a byproduct, we also give explicit forms of corrections in the respective change of variable formulas

Multidimensional SDEs with singular drift and universal construction of the polymer measure with white noise potential

We study existence and uniqueness of solution for stochastic differential equations with distributional drift by giving a meaning to the Stroock-Varadhan martingale problem associated such equations. The approach we exploit is the one of paracontrolled distributions introduced in [13]. As a result we make sense of the three dimensional polymer measure with white noise potential.Comment: We improved the presentation, corrected some of the proofs and added the global existence for the polymer measure in dimension

La confiance du consommateur vis-à-vis d’un marchand Internet : proposition d’une échelle de mesure.

L’objectif de cet article est de développer une échelle de mesure de la confiance du consommateur vis-à-vis d’un marchand Internet. Les données ont été collectées sur un échantillon total de 430 personnes. Deux études ont été réalisées pour construire cet outil de mesure. Les résultats de la recherche attestent de la fiabilité et de la validité de l’échelle.This paper develops a scale to measure consumer trust towards a web merchant. For this purpose, two studies were carried out. A total of 430 valid observations were used for the scale development. Exploratory and confirmatory factor analyses were performed to assess the psychometric properties of the items. Evidence of the scale dimensionality, reliability and validity are provided.Commerce électronique; Marchand internet; Confiance du consommateur; E-commerce; Web merchant; Consumer confidence;