# Derivative Formulas in Measure on Riemannian Manifolds

## TL;DR

This paper explores the relationships between different notions of derivatives for functions on the space of finite measures over a Riemannian manifold, providing formulas to compute intrinsic and $L$-derivatives.

## Contribution

It characterizes the links between extrinsic, intrinsic, and linear functional derivatives on measure spaces over Riemannian manifolds, simplifying their calculation.

## Key findings

- Extrinsic derivative equals linear functional derivative for certain functions.
- Intrinsic derivative matches the $L$-derivative under specified conditions.
- Provides explicit formulas for derivatives on measure spaces, extendable to probability measures.

## 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 $D^Ef$ coincides with the linear functional derivative $D^Ff$, the intrinsic derivative $D^If$ equals to the $L$-derivative $D^Lf$, and $$D^If(\eta)(x)= D^{L}f(\eta)(x)= \lim_{s\downarrow 0} \frac 1 s \nabla f(\eta+s \delta_\cdot)(x) = \nabla \big\{D^E f (\eta)\big\}(x), \ \ (x,\eta)\in M\times\mathbb M,$$ where $\nabla$ is the gradient on $M$, $\delta_x$ is the Dirac measure at $x$, and $$D^Ef(\eta)(x):= \lim\limits_{s\downarrow 0} \frac { f(\eta+s \delta_x)-f(\eta)} s,\ \ x\in M$$ 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.