Derivative Formulas in Measure on Riemannian Manifolds

Abstract

We characterise the link of derivatives in measure, which are introduced in [AKR,Card,ORS] respectively by different means, for functions on the space $\mathbb{M}$ 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 $$D^{I}f(\eta )(x)=D^{L}f(\eta )(x)=\underset{s\downarrow 0}{\lim }\frac{1}{s}\nabla f(\eta +s{\delta }_{\cdot })(x)=\nabla \{D^{E}f(\eta )\}(x),(x,\eta )\in M\times \mathbb{M},$$ where $\nabla$ is the gradient on $M$, ${\delta }_x$ is the Dirac measure at $x$, and $$D^{E}f(\eta )(x):=\underset{s\downarrow 0}{\lim }\frac{f(\eta +s{\delta }_x)-f(\eta )}{s},x\in M$$ is the extrinsic derivative of $f$ at $\eta \in \mathbb{M}$. This gives a simple way to calculate the intrinsic or $L$-derivative, and is extended to functions of probability measures. %This provides a simple way to calculate the intrinsic/Lions derivative.