  The objective of this article is to prove existence and weak uniqueness of a
Walsh spider diffusion process, whose spinning measure and coefficients are
allowed to depend on the local time spent at the junction vertex. The
methodology is to show carefully that an effectively designed martingale
problem is well-posed. Exploiting fully the results coming from the pioneering
work of [16], the construction of the solution is performed using a
concatenation procedure, as introduced in the seminal reference [26].
Uniqueness is shown by making use of the recent results obtained in [25] for
the solution of the corresponding parabolic PDE that involves a new class of
transmission condition called local time Kirchhoff 's transmission condition.
As a byproduct of our main result, we manage to compute the explicit law of the
diffusion when it behaves as a standard Brownian motion on each branch. The
case I = 2 permits us to derive also that there is existence and uniqueness for
solutions of generalized SDE on the real line that involve the local time of
the unknown process in all its coefficients.
