Projective Identification of Diffusions and Convex Geometry

This paper studies the identification of diffusion operators on the circle from positive projectiveclasses of their responses to prescribed probe functions. Since each response is observed only up toan independent unknown positive multiplicative factor, the reconstruction requires both coefficientidentification and relative-gain calibration. For general smooth uniformly elliptic operators, it isshown that three canonical probes determine the drift and diffusion coefficients up to a commonpositive scale. The main result establishes that, within the Brownian-compatible class, two suitablychosen probes suffice under a pointwise nondegeneracy condition. The Brownian relation betweenthe drift and diffusion coefficients provides a differential compatibility identity that determines therelative gain and enables explicit coefficient recovery. The recovered diffusion coefficient is thenconnected to the geometry of a smooth strictly convex planar curve: its normalized inverse squareroot determines the normalized curvature-radius profile, from which the curve is reconstructedthrough its support function, up to translation and scale. Local stability estimates are derived forgain calibration, coefficient recovery, and geometric reconstruction. Finally, a constrained weightedrelaxation is introduced for admissible profiles, and an explicit exponential contraction estimate isestablished. These results provide a projective framework for identifying diffusion operators andreconstructing normalized convex geometry under Brownian compatibility