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.
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