  Inspired by the collective phenomenon of territorial emergence, whereby
animals move and interact through the scent marks they deposit, we study the
dynamics of a 1D Brownian walker in a random environment consisting of
confining boundaries that are themselves diffusing anomalously. We show how to
reduce, in certain parameter regimes, the non-Markovian, many-body problem of
territoriality to the analytically tractable one-body problem studied here. The
mean square displacement (MSD) of the 1D Brownian walker within subdiffusing
boundaries is calculated exactly and generalizes well known results when the
boundaries are immobile. Furthermore, under certain conditions, if the boundary
dynamics are strongly subdiffusive, we show the appearance of an interesting
non-monotonicity in the time dependence of the MSD, giving rise to transient
negative diffusion.
