Transportation Distance between Probability Measures on the Infinite Regular Tree

In the infinite regular tree $\mathbb{T}_{q+1}$ with $q \in \mathbb{Z}_{\ge 2}$, we consider families $\{\mu_u^n\}$, indexed by vertices $u$ and nonnegative integers ("discrete time steps") $n$, of probability measures such that $\mu_u^n(v) = \mu_{u'}^n(v')$ if the distances $\operatorname{dist}(u,v)$ and $\operatorname{dist}(u',v')$ are equal. Let $d$ be a positive integer, and let $X$ and $Y$ be two vertices in the tree which are at distance $d$ apart. We compute a formula for the transportation distance $W_1\!\left( \mu_X^n, \mu_Y^n \right)$ in terms of generating functions. In the special case where $\mu_u^n = \mathfrak{m}_u^n$ are measures from simple random walks after $n$ time steps, we establish the linear asymptotic formula $W_1\!\left( \mathfrak{m}_X^n, \mathfrak{m}_Y^n \right) = An + B + o(1)$, as $n \to \infty$, and give the formulas for the coefficients $A$ and $B$ in closed forms. We also obtain linear asymptotic formulas in the cases of spheres and uniform balls as the radii tend to infinity. We show that these six coefficients (two from simple random walks, two from spheres, and two from uniform balls) are related by inequalities.