We define a local time flow of skew Brownian motions ,that is, a family of solutions to the stochastic differential equation defining the skew Brownian motion, starting from different points but driven by the same Brownian motion. We prove several results on distributional and path properties of the flow. Our main result is a version of the Ray–Knight theorem on local times. In our case, however, the local time process viewed as a function of the spatial variable is a pure jump Markov process rather than a diffusion.
"Local time flow related to skew brownian motion." Ann. Probab. 29 (4) 1693 - 1715, October 2001. https://doi.org/10.1214/aop/1015345768