A Temporal Spatial Minimax Rate for Smoothly-Varying Distributions in Wasserstein Space

We study the minimax rate of estimating a future value μtn+h of a curve tmapstoμt in the 2-Wasserstein space P2(ℝd) from finitely many noisy snapshots of its past, under an adiabatic bound |nablatk v|≤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 W2-risk with M-exponent γd(k+1)/(k+1+γd), γd=min(1/d,1/2) (M the total sample size). It follows from a temporal-to-spatial reduction: the smoothness budget defines a reachable W2-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 hk+1 -- the irreducible cost of an unobserved future, present even with the exact past -- and the spatial estimation curse M-γd, recovering the static distribution-estimation rate as k→∞. 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≥3) and, in a translation submodel, for all k; for k≥1 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.