  Inspired by allocation strategies in multi-armed bandit model, we propose a
pathwise construction of Walsh diffusions. For any infinitesimal generator on a
star shaped graph, there exists a unique time change associated with a
multi-parameter process such that the time change of this multi-parameter
process is the desired diffusion. The time change has an interpretation of time
allocation of the process on each edge, and it can be derived explicitly from a
family of equations.
