  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.
