We investigate the entropy $H(\mu,t)$ of a probability measure $\mu$ along the heat flow and more precisely we seek for closed algebraic representations of its derivatives. Provided that $\mu$ admits moments of any order, it is indeed proved in [Guo et al., 2010] that $t\mapsto H(\mu,t)$ is smooth, and in [Ledoux, 2016] that its derivatives at zero can be expressed into multivariate polynomials evaluated in the moments (or cumulants) of $\mu$. In the seminal contribution \cite{Led}, these algebraic expressions are derived through $\Gamma$-calculus techniques which provide implicit recursive formulas for these polynomials. Our main contribution consists in a fine combinatorial analysis of these inductive relations and for the first time to derive closed formulas for the leading coefficients of these polynomials expressions. Building upon these explicit formulas we revisit the so-called "MMSE conjecture" from [Guo et al., 2010] which asserts that two distributions on the real line with the same entropy along the heat flow must coincide up to translation and symmetry. Our approach enables us to provide new conditions on the source distributions ensuring that the MMSE conjecture holds and to refine several criteria proved in [Ledoux, 2016]. As illustrating examples, our findings cover the cases of uniform and Rademacher distributions, for which previous results in the literature were inapplicable.

翻译：我们研究了概率测度$\mu$沿热流的熵$H(\mu,t)$，并更精确地寻求其导数的闭代数表示。若$\mu$具有任意阶矩，文献[Guo et al., 2010]已证明$t\mapsto H(\mu,t)$是光滑的，而[Ledoux, 2016]证明其在零点的导数可表示为以$\mu$的矩（或累积量）为变量的多元多项式。在开创性文献\cite{Led}中，这些代数表达式通过$\Gamma$-演算技术导出，为该类多项式提供了隐式的递归公式。我们的主要贡献在于对这些递推关系进行精细的组合分析，并首次推导出这些多项式表达式首项系数的闭式公式。基于这些显式公式，我们重新审视[Guo et al., 2010]中提出的"MMSE猜想"——该猜想断言实轴上沿热流具有相同熵的两个分布必然在平移和对称变换下重合。我们的方法能够为源分布提供确保MMSE猜想成立的新条件，并改进[Ledoux, 2016]中证明的若干判别准则。作为示例，我们的研究结果涵盖了均匀分布与Rademacher分布的情形，而先前文献中的结论对此类情形并不适用。