Embedding of Walsh Brownian motion

Let (Z,κ) be a Walsh Brownian motion with spinning measure κ. Suppose μ is a probability measure on Rⁿ. We first provide a necessary and sufficient condition for μ to be a stopping distribution of (Z,κ). Then if the stopped process is required to be uniformly integrable, we show that such a stopping time exists if and only if μ is balanced. Next, under the assumption of being balanced, we identify the minimal stopping times with those τ such that the stopped process Z^τ is uniformly integrable. Finally, we generalize Vallois’ embedding, and prove that it minimizes the expectation E[Ψ(L_τ^Z)] among all the admissible solutions τ, where Ψ is a strictly convex function and (L_t^Z)_t≥0 is the local time of the Walsh Brownian motion at the origin.