We study the minimax rate of estimating a future value $μ_{t_n+h}$ of a curve $t\mapstoμ_t$ in the $2$-Wasserstein space $\mathcal{P}_2(\mathbb{R}^d)$ from finitely many noisy snapshots of its past, under an adiabatic bound $\|\nabla_t^k v\|\le\varepsilon$ on the $k$-th covariant derivative of the velocity field. Our central result is a unified temporal-spatial minimax lower bound: over regular, locally transport-rich subclasses, every estimator incurs $W_2$-risk with $M$-exponent $γ_d(k+1)/(k+1+γ_d)$, $γ_d=\min(1/d,1/2)$ ($M$ the total sample size). It follows from a temporal-to-spatial reduction: the smoothness budget defines a reachable $W_2$-ball into which a transport packing is embedded along the time axis, and the information of the entire snapshot experiment is controlled by a Fano argument -- the spatial packing is classical, but its smoothness-admissible temporal embedding and the full-window analysis are new. The bound interpolates a dimension-free extrapolation floor of order $\varepsilon h^{k+1}$ -- the irreducible cost of an unobserved future, present even with the exact past -- and the spatial estimation curse $M^{-γ_d}$, recovering the static distribution-estimation rate as $k\to\infty$. We state the lower bound in a design-dependent form -- with a design-weighted effective sample size -- valid for arbitrary observation times, and obtain the closed-form exponent in the dense (equispaced) regime. The matching upper bound is established at $k=0$ (rate $M^{-1/(d+1)}$, $d\ge3$) and, in a translation submodel, for all $k$; for $k\ge1$ a covariant estimator attains the rate conditionally on two estimates (a comparison-geometry bias bound and an optimal-transport map-estimation rate), leaving the unconditional general-$k$ upper bound as an open problem. Numerical experiments on synthetic curved and flat families corroborate the predicted exponents.

翻译：我们研究在$2$-Wasserstein空间$\mathcal{P}_2(\mathbb{R}^d)$中，从有限个带噪快照估计曲线$t\mapsto\mu_t$的未来值$\mu_{t_n+h}$的极小极大速率，其中速度场的第$k$阶协变导数满足绝热界$\|\nabla_t^k v\|\le\varepsilon$。核心结果是一个统一的时间-空间极小极大下界：在正则的、局部传输丰富的子类上，每个估计量的$W_2$-风险具有$M$指数$\gamma_d(k+1)/(k+1+\gamma_d)$，其中$\gamma_d=\min(1/d,1/2)$（$M$为总样本量）。该下界源于时间到空间的约化：光滑性预算定义了可达的$W_2$-球，沿时间轴嵌入传输填充，且整个快照实验的信息由Fano论证控制——空间填充是经典的，但其光滑可容许的时间嵌入和全窗口分析是新的。该界插值了维数无关的外推下界$\varepsilon h^{k+1}$（即使已知精确过去，未观测未来的不可约代价）和空间估计的维数诅咒$M^{-\gamma_d}$，当$k\to\infty$时恢复静态分布估计速率。我们以设计依赖形式表述该下界——包含设计加权的有效样本量——适用于任意观测时间，并在密集（等间距）情形下获得封闭形式的指数。当$k=0$时（速率为$M^{-1/(d+1)}$，$d\ge3$），以及在平移子模型中对所有$k$，建立了匹配的上界；对于$k\ge1$，协变估计量在两个估计（比较几何偏差界和最优传输映射估计速率）的条件依赖下达到该速率，将无条件的一般$k$上界留作未解决问题。在合成弯曲和平坦族上的数值实验验证了预测的指数。