Time homogeneous diffusion with drift and killing to meet a given marginal

In this article, it is proved that for any probability law μ over R and a drift field b:R→R and killing field k:R→R+ which satisfy hypotheses stated in the article and a given terminal time t>0, there exists a string m, an α∈(0,1], an initial condition x0∈R and a process X with infinitesimal generator (12∂2∂m∂x+b∂∂m−∂K∂m) where k=∂K∂x such that for any Borel set B∈B(R), P(Xt∈B|X0=x0)=αμ(B). Firstly, it is shown the problem with drift and without killing can be accommodated, after a simple co-ordinate change, entirely by the proof in Noble (2013). The killing field presents additional problems and the proofs follow the lines of Noble (2013) with additional arguments.