A Variational Analysis of Precision-Weighted Inference on Wasserstein Space

We study a class of precision-weighted variational problems posed on the Wasserstein space of probability measures. The objective functional combines relative entropy with respect to a reference measure and a quadratic data-fidelity term weighted by a nonnegative precision parameter. The parameter controls the strength of enforcement of an observational constraint expressed through expectation operators acting on the measure. Under structural assumptions including lower semicontinuity, coercivity, and compactness of sublevel sets in the topology of narrow convergence with uniform moment bounds, existence of minimizers is established via the direct method in the calculus of variations. We identify a displacement-convex regime in which geodesic convexity of both entropy and the data-fidelity term implies convexity of the full functional and uniqueness of minimizers. We emphasize that increasing precision may destroy geodesic convexity, thereby invalidating convexity-based uniqueness arguments without implying nonexistence. The large-precision limit is analyzed through Gamma-convergence. As the precision parameter tends to infinity, the penalized functional converges variationally to a constrained entropy minimization problem in which the observational constraint is enforced exactly. Subsequence convergence of minimizers is established under compactness assumptions, while no rate or selection result is asserted beyond the Gamma-limit structure. Beyond static minimization, we construct generalized Wasserstein gradient-flow trajectories using a Jordan–Kinderlehrer–Otto minimizing-movement scheme. Existence of discrete-time minimizers follows from lower semicontinuity and compactness. Discrete energy dissipation and quantitative bounds on constraint violation are derived, showing that large precision suppresses constraint deviations at order 1 over precision. We further introduce a regularized particle approximation at the level of empirical measures and formulate the corresponding discrete minimizing-movement scheme, isolating structural consistency without asserting propagation-of-chaos rates. In the high-precision regime, we analyze variational barrier structure between distinct minimizers and show that barrier heights scale at least linearly with the precision parameter under suitable separation hypotheses. Under a conditional path-space large deviation framework, we define the associated quasipotential and explain how increasing precision amplifies transition costs while small noise and large particle number suppress rare transitions. The paper provides a unified variational, dynamical, and asymptotic framework for precision-weighted inference on Wasserstein space and clarifies the structural mechanisms governing constraint collapse, convexity breakdown, and metastable behavior in high-precision regimes.