  We measure the thermalization dynamics of a lattice Bose gas that is Stark
localized by a parabolic potential. A non-equilibrium thermal density
distribution is created by quickly removing an optical barrier. The resulting
spatio-temporal dynamics are resolved using Mardia's $B$ statistic, which is a
measure sensitive to the shape of the entire density distribution. We conclude
that equilibrium is achieved for all lattice potential depths that we sample,
including the strongly interacting and localized regime. However,
thermalization is slow and non-exponential, requiring up to 500 tunneling
times. We show that the Hubbard $U$ term is not responsible for thermalization
via comparison to an exact diagonalization calculation, and we rule out
equilibration driven by lattice-light heating by varying the laser wavelength.
The thermalization timescale is comparable to the next-nearest-neighbor
tunneling time, which suggests that a continuum, strongly interacting theory
may be needed to understand equlibration in this system.
