  For a holographically defined finite temperature theory, we derive an
off-shell constitutive relation for a global $U(1)$ current driven by a weak
external non-dynamical electromagnetic field. The constitutive relation
involves an all order gradient expansion resummed into three momenta-dependent
transport coefficient functions: diffusion, electric conductivity, and
"magnetic" conductivity. These transport functions are first computed
analytically in the hydrodynamic limit, up to third order in the derivative
expansion, and then numerically for generic values of momenta. We also compute
a diffusion memory function, which, as a result of all order gradient
resummation, is found to be causal.
