Feller's One-Dimensional Diffusions as Weak Solutions to Stochastic Differential Equations

A continuous strong Markov process X on the line generated by Feller's generalized second order differential operator D_mD is considered. Supposed that the canonical scale p is locally the difference of two bounded convex functions, that the speed measure m contains a strictly positive absolutely continuous component, and that both boundaries of the state space R are inaccessible. Then the process X is characterized as a weak solution to a stochastic differential equation involving local time.