Regularity and Pathwise bounds for probabilistic solutions of PDEs
Mouhamadou Sy
TL;DR
The paper addresses transforming ensemble (probabilistic) bounds into pathwise controls for PDE solutions. It introduces an estimation procedure that combines a local increment condition with a probabilistic ensemble bound, enabling almost-sure trajectory bounds. The method yields concrete pathwise growth estimates for nonlinear Schrödinger equations: a $3D$ cubic NLS has a pathwise $H^2$ bound of order $(1+t)^{11/12}$ and a energy-supercritical NLS in $H^s$ gains a bound of order $(1+t)^{1/2}$ modulated by a slowly varying function, namely $(\xi(1+\sqrt{\ln(1+t)}))^{p-1}$. These results provide a framework to analyze long-time behavior and global well-posedness through trajectorywise controls, extending probabilistic global well-posedness results to individual-time evolution estimates with explicit growth rates.
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.