Transition density of one-dimensional diffusion with discontinuous drift

The paper studies the transition density of a one-dimensional diffusion process with a discontinuous drift coefficient. A probabilistic representation of the transition density is given, which illustrates the close connections between the discontinuities of the drift and Brownian local times. In addition, some explicit results are obtained based upon the trivariate density of Brownian motion, its occupation, and local times. The transition density of a bang-bang control example is obtained explicitly.