Existence of a reference measure for measure-valued derivations

Determine whether every measure-valued derivation on a complete separable metric space admits a finite reference measure that represents it as a derivation with values in an associated $L^\infty$ space.

Background

The paper defines a measure-valued derivation as a linear map from $Lip(X)$ into finite signed measures on space-time that satisfies the Leibniz rule. A finite-mass measure-valued derivation is one for which the measures generated by Lipschitz functions are dominated by a common finite measure weighted by the asymptotic Lipschitz constant.

For a derivation represented by a pair $(V,\lambda)$, the action has the form $\mathcal D(f)=Vf\,\lambda$, where $V$ is an $L^\infty(\lambda)$-valued derivation and $\lambda$ is a reference measure. The authors establish existence of such a representation for finite-mass measure-valued derivations by constructing the minimal mass measure and applying $L^1$–$L^\infty$ duality. The explicitly unresolved issue is whether an arbitrary measure-valued derivation, without the finite-mass assumption, necessarily possesses any reference measure representation.