To certain non-linear Schrödinger (NLS) equations, one can associate an invariant Gibbs measure based on the conserved energy. This is the basic ingredient of the Euclidean approach to constructive quantum field theory, as well as the large-time asymptote for the stochastic non-linear heat equation. I shall discuss a certain mean-field limit relating these Gibbs measures and the positive-temperature equilibria of the underlying many-body model (the Bose gas). A difficulty is that the Gibbs measure lives on low regularity distributional spaces, so that the non-linearity has to be understood in a renormalized sense. I shall put emphasis on the control of the renormalization procedure at the level of the quantum many-body model. joint work with Mathieu Lewin and Phan Thành Nam.