Derivative formulas in measure on Riemannian manifolds

Abstract

We characterize the link of derivatives in measure, which are introduced in Albeverio, Kondratiev, and Röckner (C. R. Acad. Sci. Paris Sér I Math. 323 (1996) 1129–1134); Cardaliaguet (P.-L. Lions lectures at College de France. Online at https://www.ceremade.dauphine.fr/~cardaliaguet/MFG20130420.pdf); and Overbeck, Röckner, and Schmuland (Ann. Probab. 23 (1995) 1–36), respectively, by different means, for functions on the space M of finite measures over a Riemannian manifold M. For a reasonable class of functions f, the extrinsic derivative D E f coincides with the linear functional derivative D F f , the intrinsic derivative D I f equals to the L-derivative D L f , and D I f ( η ) ( x ) = D L f ( η ) ( x ) = lim s ↓ 0 1 s ∇ f ( η + s δ · ) ( x ) = ∇ { D E f ( η ) } ( x ) , ( x , η ) ∈ M × M , where ∇ is the gradient on M, δ x is the Dirac measure at x, and D E f ( η ) ( x ) : = lim s ↓ 0 f ( η + s δ x ) − f ( η ) s , x ∈ M is the extrinsic derivative of f at η ∈ M . This gives a simple way to calculate the intrinsic or L-derivative, and is extended to functions of probability measures.