TRANSPORTATION DISTANCE BETWEEN PROBABILITY MEASURES ON THE INFINITE REGULAR TREE

In the infinite regular tree \BbbTq+1 with q \in \BbbZ_\geq2, we consider families \{\muⁿ_u\}, indexed by vertices u and nonnegative integers (``discrete time steps"") n, of probability measures such that \muⁿ_u(v) = \muⁿ_u\prime(v^\prime) if the distances d(u, v) and d(u^\prime, v^\prime) 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 W1^\bigl(\muⁿ_X, \muⁿ_Y^\bigr) in terms of generating functions. In the special case where \muⁿ_u = \frakmⁿ_u are measures from simple random walks after n time steps, we establish the linear asymptotic formula W1^\bigl(\frakmⁿ_X, \frakmⁿ_Y^\bigr) = An + B + o(1), as n \rightarrow \infty, and give the formulas for the coefficients A and B in closed forms. We also obtain linear asymptotic formulas when \muⁿ_u is the uniform distribution on the sphere or on the ball of radius n as n \rightarrow \infty. We show that these six coefficients (two from the simple random walk, two from the uniform distribution on the sphere, and two from the uniform distribution on the ball) are related by inequalities.