Gibbs measures for self-interacting Wiener paths

In this note we study a class of specifications over $d$-dimensional Wiener measure which are invariant under uniform translation of the paths. This degeneracy is removed by restricting the measure to the $\sigma$-algebra generated by the increments of the coordinate process. We address the problem of existence and uniqueness of Gibbs measures and prove a central limit theorem for the rescaled increments. These results apply to the study of the ground state of the Nelson model of a quantum particle interacting with a scalar boson field.Comment: 15 pages, no figures; typos, details added to the proof

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

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