In various disparate settings one studies how random processes give way to parabolic partial differential equations and in turn to functional inequalities involving gradients and variational calculus. Particles subject to gradient forces, the fields which move them, the empirical measures which they form, the smooth laws which those measures approximate, and differentiable functionals of these laws constitute a hierarchy of descriptions of diffusion. Intervening on this hierarchy is a choice of how a conintuity operator converts particle velocity to the evolution of measures. Here a central object is constructed mediating two different such descriptions. The space of smooth positive probability densities on a closed Riemannian manifold is treated as a Fréchet manifold whose full continuous cotangent space consists of nonconstant distributions and whose regular cotangent space consists of sufficiently regular nonconstant functions. Beginning from the operator on this space arising as the infinitesimal lift of diffeomorphisms of the base manifold, two different Hilbert completions of this space of densities account for a wide class of objects relevant, leading to a partial universalisation of the microscopic--mesoscopic--macroscopic hierarchy in Markov analysis.
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