Stability for the Bakry-Emery theorem

The Bakry-Emery theorem states that if a probability measure is in some sense more log-concave than the standard Gaussian measure, then certain functional inequalities (such as the Poincare inequality and the logarithmic Sobolev inequality) hold, with better constants than for the associated Gaussian inequalities. I will show how we can combine Stein's method and simple variational arguments to show that if the Bakry-Emery bound is almost sharp for a given measure, then that measure must almost split off a Gaussian factor, with explicit quantitative bounds. Joint work with Thomas Courtade.