# Almost sure scattering for the 4D energy-critical defocusing nonlinear wave equation with radial data

## TL;DR

This paper proves that for the 4D energy-critical defocusing nonlinear wave equation with radially symmetric initial data, solutions exist globally and scatter almost surely, even with super-critical initial data, using novel probabilistic and analytical techniques.

## Contribution

It introduces the first almost sure scattering result for an energy-critical dispersive PDE with super-critical initial data, utilizing an approximate Morawetz estimate and large deviation bounds.

## Key findings

- Almost sure global existence and scattering for randomized radial data.
- First such result for energy-critical dispersive equations with super-critical data.
- Development of new probabilistic estimates for wave evolution.

## Abstract

We consider the energy-critical defocusing nonlinear wave equation on $\mathbb{R}^4$ and establish almost sure global existence and scattering for randomized radially symmetric initial data in $H^s_x(\mathbb{R}^4) \times H^{s-1}_x(\mathbb{R}^4)$ for $\frac{1}{2} < s < 1$. This is the first almost sure scattering result for an energy-critical dispersive or hyperbolic equation with scaling super-critical initial data. The proof is based on the introduction of an approximate Morawetz estimate to the random data setting and new large deviation estimates for the free wave evolution of randomized radially symmetric data.