Regularity and Pathwise bounds for probabilistic solutions of PDEs

TL;DR

This paper introduces a new method to derive individual, pathwise bounds for solutions of PDEs from ensemble bounds, enhancing understanding of long-term behavior, with applications to nonlinear Schrödinger equations.

Contribution

It develops a flexible estimation procedure that transforms ensemble bounds into pathwise controls, differing from Bourgain's local wellposedness approach.

Findings

Provides new pathwise bounds for NLS solutions

Transforms ensemble bounds into individual solution controls

Enhances understanding of long-time solution behavior

Abstract

In this paper, we build a procedure that allows to establish regularity and controls in time for probabilistic solutions to PDEs. Probabilistic approaches to global wellposedness problems usually provide ensemble bounds on the solutions. These bounds are the main tools to ensure convergence procedures yielding the existence and uniqueness of global solutions. A question of interest consists in transforming such ensemble bounds into individual controls on the flow ; this, among other uses, gives valuable information on the long-time behavior of the solutions. Toward such question of bounds transformation, Bourgain initiated a successful procedure that exploited the local wellposedness of the PDE, with an estimate of the time of size-doubling. In this note, we construct an estimation procedure which relies on a different local requirement. It turns out that this substitute is flexible enough to be possible to fulfill with the help of the ensemble bound itself. For applications of the procedure, we are able to provide new pathwise controls on solutions to NLS equations.