We start this talk by recalling briefly the last results I have obtained on Walsh spider diffusion having a spinning measure selected from the own local time spent by the process at the vertex. We emphasize the key "non-stickiness" property of the process at the junction point, which allows to eliminate all the discontinuities of the corresponding operator at the vertex in the Itô's rule. Thereafter, we will explain how the "non-stickiness" condition has given us the main intuition to build test functions for our comparison theorem, solutions of ODE with well designed coefficients. Moreover, we will see that the key point is to impose a "local-time" derivative at  the vertex absorbing the error term induced by the Kirchhoff's speed of the Hamiltonians. Finally, we will discuss the extension of the latest results to a more general Kirchhoff-spinning measure, and how an associated corresponding master equation will completely open a new wonderful problem in the field of stochastic control theory.