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Given an initial (resp., terminal) probability measure $\mu$ (resp., $\nu$)on $\mathbb{R}^d$, we characterize those optimal stopping times $\tau$ thatmaximize or minimize the functional $\mathbb{E} |B_0 - B_\tau|^{\alpha}$,$\alpha > 0$, where $(B_t)_t$ is Brownian motion with initial law $B_0\sim \mu$and with final distribution --once stopped at $\tau$-- equal to $B_\tau\sim\nu$.  The existence of such stopping times is guaranteed by Skorohod-typeembeddings of probability measures in "subharmoic order" into Brownian motion.This problem is equivalent to an optimal mass transport problem with certainconstraints, namely the optimal subharmonic martingale transport. Under theassumption of radial symmetry on $\mu$ and $\nu$, we show that the optimalstopping time is a hitting time of a suitable barrier, hence is non-randomizedand is unique.

Optimal Brownian Stopping between radially symmetric marginals in general dimensions