  Whether in the thermodynamic limit of lattice length infinite, hole
concentration tending to zero, nonzero temperature, and U/t > 0 the charge
stiffness of the 1D Hubbard model with first neighbor transfer integral t and
on-site repulsion U is finite or vanishes and thus whether there is or there is
no ballistic charge transport, respectively, remains an unsolved and
controversial issue, as different approaches yield contradictory results. In
this paper we provide an upper bound on the charge stiffness and show that
(similarly as at zero temperature), for T >0 and U/t>0 it vanishes in the limit
of zero hole concentration within the canonical ensemble in the thermodynamic
limit. Moreover, we show that at high temperature the charge stiffness vanishes
as well within the grand-canonical ensemble in the infinite lattice length
limit and chemical potential approaching half the Mott-Hubbard gap. The lack of
charge ballistic transport indicates that charge transport at finite
temperatures is dominated by a diffusive contribution. Our scheme uses a
suitable exact representation of the electrons in terms of rotated electrons
for which the numbers of singly occupied and doubly occupied lattice sites are
good quantum numbers for U/t>0. In contrast to often less controllable
numerical studies, the use of such a representation reveals the carriers that
couple to the charge probes and provides useful physical information on the
microscopic processes behind the exotic charge transport properties of the 1D
electronic correlated system under study.
