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

Walsh Diffusions as Time Changed Multi-parameter Processes