Probabilistic small data global well-posedness of the energy-critical Maxwell-Klein-Gordon equation

TL;DR

This paper proves global well-posedness for the energy-critical Maxwell-Klein-Gordon equation with super-critical random initial data using probabilistic methods, marking a first in geometric wave equations.

Contribution

It introduces a novel probabilistic approach and a parametrix construction to establish global existence at super-critical regularity for a geometric wave equation.

Findings

First global existence result for a geometric wave equation with super-critical random data.

Uses an induction on frequency and probabilistic parametrix construction.

Achieves well-posedness relative to Coulomb gauge.

Abstract

We establish probabilistic small data global well-posedness of the energy-critical Maxwell-Klein-Gordon equation relative to the Coulomb gauge for scaling super-critical random initial data. The proof relies on an induction on frequency procedure and a modified linear-nonlinear decomposition furnished by a delicate "probabilistic" parametrix construction. This is the first global existence result for a geometric wave equation for random initial data at scaling super-critical regularity.