Simulating a diffusion on a graph. Application to reservoir engineering

We develop a simple Monte Carlo method to compute the position at a given time of a diffusion on a graph, with constant speed on each edge. This method is exact, and we claim it could be used for simulating the position of a particle in a fissured media. Besides, we advocate that the notion of diffusions on graphs could be useful to understand the behavior of one-dimensional diffusions whose infinitesimal generator has piecewise constant coefficients.