Embedding of Walsh Brownian Motion

Let $(Z,\kappa)$ be a Walsh Brownian motion with spinning measure $\kappa$. Suppose $\mu$ is a probability measure on $\mathbb{R}^n$. We characterize all the $\kappa$ such that $\mu$ is a stopping distribution of $(Z,\kappa)$. If we further restrict the solution to be integrable, we show that there would be only one choice of $\kappa$. We also generalize Vallois' embedding, and prove that it minimizes the expectation $\mathbb{E}[\Psi(L^Z_{\tau})]$ among all the admissible solutions $\tau$, where $\Psi$ is a strictly convex function and $(L_t^Z)_{t \geq 0}$ is the local time of the Walsh Brownian motion at the origin.