  We introduce a location statistic for distributions on non-linear geometric
spaces, the diffusion mean, serving as an extension and an alternative to the
Fr\'echet mean. The diffusion mean arises as the generalization of Gaussian
maximum likelihood analysis to non-linear spaces by maximizing the likelihood
of a Brownian motion. The diffusion mean depends on a time parameter $t$, which
admits the interpretation of the allowed variance of the diffusion. The
diffusion $t$-mean of a distribution $X$ is the most likely origin of a
Brownian motion at time $t$, given the end-point distribution $X$. We give a
detailed description of the asymptotic behavior of the diffusion estimator and
provide sufficient conditions for the diffusion estimator to be strongly
consistent. Particularly, we present a smeary central limit theorem for
diffusion means and we show that joint estimation of the mean and diffusion
variance rules out smeariness in all directions simultaneously in general
situations. Furthermore, we investigate properties of the diffusion mean for
distributions on the sphere $\mathbb S^n$. Experimentally, we consider
simulated data and data from magnetic pole reversals, all indicating similar or
improved convergence rate compared to the Fr\'echet mean. Here, we additionally
estimate $t$ and consider its effects on smeariness and uniqueness of the
diffusion mean for distributions on the sphere.
