  We study finite-temperature magnetization transport in a one-dimensional
anisotropic Heisenberg model, focusing in particular on the gapped phase. Using
numerical simulations by two different methods, a propagation of localized
wavepackets and a study of nonequilibrium steady states of a master equation in
a linear-response regime, we conclude that the transport at finite temperatures
is diffusive. With decreasing temperature the diffusion constant increases,
possibly exponentially fast. This means that at low temperatures the transition
from ballistic to asymptotic diffusive behavior happens at very long times. We
also study dynamics of initial domain wall like states, showing that on the
attainable time scales they remain localized.
