# Invariance of white noise for KdV on the line

## TL;DR

This paper proves that solutions to the KdV equation with white noise initial data on the real line exist almost surely, preserve white noise distribution over time, and satisfy the group property, also providing new insights for the torus case.

## Contribution

It establishes the invariance of white noise for KdV on the line and offers a novel proof for the torus setting, advancing understanding of stochastic PDE invariance.

## Key findings

- Solutions exist almost surely with white noise initial data.
- Solutions obey the group property over time.
- White noise law is preserved at all times.

## Abstract

We consider the Korteweg--de Vries equation with white noise initial data, posed on the whole real line, and prove the almost sure existence of solutions. Moreover, we show that the solutions obey the group property and follow a white noise law at all times, past or future. As an offshoot of our methods, we also obtain a new proof of the existence of solutions and the invariance of white noise measure in the torus setting.