We study a simple stochastic differential equation driven by one Brownian motion on a general oriented metric graph whose solutions are stochastic flows of kernels. Under some condition, we describe the laws of all solutions. This work is a natural continuation of previous works by Hajri, Hajri-Raimond and Le Jan-Raimond where some particular graphs have been considered.
"Stochastic flows on metric graphs." Electron. J. Probab. 19 1 - 20, 2014. https://doi.org/10.1214/EJP.v19-2773