Concentration of Exponential Tilts Implies Curvature

Source abstract

We show, under a smoothness assumption, that probability measures whose exponential tilts have a uniformly bounded variance are given by a strongly convex potential, up to a bounded perturbation, yielding a partial converse to the Bakry-Émery theorem combined with Holley-Stroock's perturbation principle. A stronger version enforcing uniform convexity of the potential is obtained by considering the low-temperature regime.