A higher-order Otto calculus approach to the Gaussian completely monotone conjecture

The Gaussian completely monotone (GCM) conjecture states that the $m$-th time-derivative of the entropy along the heat flow on $\mathbb{R}^d$ is positive for $m$ even and negative for $m$ odd. We prove the GCM conjecture for orders up to $m=5$, assuming that the initial measure is log-concave, in any dimension. Our proof differs significantly from previous approaches to the GCM conjecture: it is based on Otto calculus and on the interpretation of the heat flow as the Wasserstein gradient flow of the entropy. Crucial to our methodology is the observation that the convective derivative behaves as a flat connection over probability measures on $\mathbb{R}^d$. In particular we prove a form of the univariate Faa di Bruno's formula on the Wasserstein space (despite it being curved), and we compute the higher-order Wasserstein differentials of internal energy functionals (including the entropy), both of which are of independent interest.