440 research outputs found
Controlling Rough Paths
We formulate indefinite integration with respect to an irregular function as
an algebraic problem and provide a criterion for the existence and uniqueness
of a solution. This allows us to define a good notion of integral with respect
to irregular paths with Hoelder exponent greater than 1/3 (e.g. samples of
Brownian motion) and study the problem of the existence, uniqueness and
continuity of solution of differential equations driven by such paths. We
recover Young's theory of integration and the main results of Lyons' theory of
rough paths in Hoelder topology.Comment: 43 pages, no figures, corrected a proof in Sec.
Gibbs measures for self-interacting Wiener paths
In this note we study a class of specifications over -dimensional Wiener
measure which are invariant under uniform translation of the paths. This
degeneracy is removed by restricting the measure to the -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
Rooted trees for 3d Navier-Stokes equation
We establish a representation of a class of solutions of 3d Navier-Stokes
equations in using sums over rooted trees. We study the convergence
properties of this series recovering in a simplified manner some results
obtained recently by Sinai and other known results for solutions in spaces of
pseudo-measures introduced initially by Le Jan and Sznitman. The series
representation make sense also in the critical case where there exists global
solutions for small initial data and it allows the study of their long-time or
small-distance behavior.Comment: 11 page
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
Unbounded rough drivers
We propose a theory of linear differential equations driven by unbounded
operator-valued rough signals. As an application we consider rough linear
transport equations and more general linear hyperbolic symmetric systems of
equations driven by time-dependent vector fields which are only distributions
in the time direction.Comment: 38 pages. some improvements and precision
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