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.

翻译：高斯完全单调（GCM）猜想断言：在$\mathbb{R}^d$上沿热流方向的熵的$m$阶时间导数，当$m$为偶数时为正，$m$为奇数时为负。我们证明了在任意维度下，当初始测度为对数凹时，该猜想对阶数高达$m=5$的情况成立。我们的证明方法与先前研究GCM猜想的方法有显著不同：它基于奥托演算，并将热流解释为熵的Wasserstein梯度流。我们方法的关键在于观察到对流导数在$\mathbb{R}^d$上的概率测度空间中表现为平坦联络。特别地，我们证明了Wasserstein空间（尽管是弯曲的）上的一元Faa di Bruno公式形式，并计算了内能泛函（包括熵）的高阶Wasserstein微分，这两项结果均具有独立的研究价值。