Forward and inverse problems for measure flows in Bayes Hilbert spaces

TL;DR

This paper develops a mathematical framework for analyzing forward and inverse problems involving time-dependent probability measures in Bayes--Hilbert spaces, introducing canonical dynamical realizations and regularized reconstruction methods.

Contribution

It introduces a new approach to model and reconstruct evolving probability measures in infinite-dimensional Bayes--Hilbert spaces, combining transport geometry, variational methods, and regularization techniques.

Findings

Existence of minimizers under regularity and observability assumptions

Derivation of first-variation formulas for the inverse problem

Local stability results for the data-to-solution map

Abstract

We study forward and inverse problems for time-dependent probability measures in Bayes--Hilbert spaces. On the forward side, we show that each sufficiently regular Bayes--Hilbert path admits a canonical dynamical realization: a weighted Neumann problem transforms the log-density variation into the unique gradient velocity field of minimum kinetic energy. This construction induces a transport form on Bayes--Hilbert tangent directions, which measures the dynamical cost of realizing prescribed motions, and yields a flow-matching interpretation in which the canonical velocity field is the minimum-energy execution of the prescribed path. On the inverse side, we formulate reconstruction directly on Bayes--Hilbert path space from time-dependent indirect observations. The resulting variational problem combines a data-misfit term with the transport action induced by the forward geometry. In our infinite-dimensional setting, however, this transport geometry alone does not provide sufficient compactness, so we add explicit temporal and spatial regularization to close the theory. The linearized observation operator induces a complementary observability form, which quantifies how strongly tangent directions are seen through the data. Under explicit Sobolev regularity and observability assumptions, we prove existence of minimizers, derive first-variation formulas, establish local stability of the observation map, and deduce recovery of the evolving law, its score, and its canonical velocity field under the strong topologies furnished by the compactness theory.