Stability of Poincar{\'e} constant

We study stability of the sharp Poincar{'e} constant of the invariant probability measure of a reversible diffusion process satisfying some natural conditions. The proof is based on the spectral interpretation of Poincar{'e} inequalities and Stein’s method. In particular, these results are applied to the gamma distributions and to strictly log-concave measures in dimension one, giving stability for Brascamp-Lieb inequalities.