Measure flow path recovery in Bayes Hilbert spaces

We study the ill-posed problem of recovering a probability measure flow from finitely many moving localized sensors using a Bayes Hilbert framework. Relative to a fixed reference probability measure, a probability law is represented by its centered log-ratio coordinates, so that an evolving law becomes a path in a Hilbert space of functions. For sufficiently regular Bayes Hilbert paths, we construct a canonical minimum-energy transport realization of the path by solving a weighted Neumann problem at each time, yielding an intrinsic transport form on tangent directions. We then formulate an inverse problem directly on Bayes Hilbert path space. Linearization of an observation operator yields an observability form, and recoverability is governed by its interaction with the transport geometry through a joint transport--observability form. In the ambient infinite-dimensional setting, we develop a regularized variational theory and identify limitations of localized sensing: mobile sensors can make the joint form injective, but they do not in general yield a coercive stability estimate on the full state space. This obstruction leads naturally to finite-dimensional Bayes Hilbert reductions. There the transport form becomes a kinetic tensor and the linearized observations become reduced sensing matrices, so recoverability can be expressed through explicit Gramian conditions. We show that localized bump sensors detect every fixed reduced direction, that finitely many suitably placed static sensors yield uniform reduced observability, and there exist path-dependent sensor trajectories such that even a single moving sensor can recover the reduced path. Finally, we show that these reduced recovery results lift to approximate ambient recovery for paths that are well approximated by the chosen finite-dimensional subspaces, yielding stable reconstruction up to projection error.