In this talk, we address the stability problem of the famous Brascamp-Lieb inequality for striclty log-concave probability measures on the Euclidean space. More precisely, if a given function almost satisfies the equality in the BL inequality, is it true that it is close in some sense to the underlying extremal functions ? Using a spectral interpretation of the BL inequality, we prove that the distance to the extremal functions in quadratic norm is of order square root of the deficit parameter and involves the second positive eigenvalue of a convenient diffusion operator we wish to estimate. Our results are illustrated by some examples for which the usual uniform convexity assumption on the potential is relaxed. This is a joint work with M. Bonnefont (Institut de Mathématiques de Bordeaux) and J. Serres (Institut de Mathématiques de Toulouse).