On the relation of one-dimensional diffusions on natural scale and their speed measures

TL;DR

This paper explores the topological relationship between one-dimensional diffusions on natural scale and their speed measures, establishing a homeomorphism and continuous dependence in both directions.

Contribution

It proves the converse of Stone's result, showing continuous dependence of speed measures on diffusions and characterizes their topological relation.

Findings

Homeomorphic relation between diffusions and speed measures

Continuous dependence of speed measures on diffusions

Topological characterization of the set of diffusions

Abstract

It is well-known that the law of a one-dimensional diffusion on natural scale is fully characterized by its speed measure. C. Stone proved a continuous dependence of diffusions on their speed measures. In this paper we establish the converse direction, i.e. we prove a continuous dependence of the speed measures on their diffusions. Furthermore, we take a topological point of view on the relation. More precisely, for suitable topologies, we establish a homeomorphic relation between the set of regular diffusions on natural scale without absorbing boundaries and the set of locally finite speed measures.