# Almost sure global well posedness for the BBM equation with infinite $L^{2}$ initial data

## TL;DR

This paper proves that the Benjamin-Bona-Mahony (BBM) equation is almost surely globally well-posed for initial data with infinite $L^{2}$ regularity on the one-dimensional torus, using probabilistic methods and the $I$-method.

## Contribution

It introduces a probabilistic approach to establish global well-posedness of BBM with initial data below $L^{2}$, extending the understanding of ill-posed regimes.

## Key findings

- Almost sure global well-posedness for initial data below $L^{2}$.
- Use of the $I$-method to control solution growth.
- Discussion on stability in the ill-posed regime.

## Abstract

We consider the probabilistic Cauchy problem for the Benjamin-Bona-Mahony equation (BBM) on the one-dimensional torus $\mathbb{T}$ with initial data below $L^{2}(\mathbb{T})$. With respect to random initial data of strictly negative Sobolev regularity, we prove that BBM is almost surely globally well-posed. The argument employs the $I$-method to obtain an a priori bound on the growth of the `residual' part of the solution. We then discuss the stability properties of the solution map in the deterministically ill-posed regime.