  This paper is devoted to Gaussian interpolation inequalities with endpoint
cases corresponding to the Gaussian Poincar\'e and the logarithmic Sobolev
inequalities, seen as limits in large dimensions of Gagliardo-Nirenberg-Sobolev
inequalities on spheres. Entropy methods are investigated using not only heat
flow techniques but also nonlinear diffusion equations as on spheres. A new
stability result is established for the Gaussian measure, which is directly
inspired by recent results for spheres.
