By performing an elementary transformation, the conventional velocity autocorrelation function expression for the temperature and density dependent self-diffusion constant D(T,p) has been reformulated to emphasize how initial particle momentum biases final mean displacement. Using collective flow variables, an analogous expression has been derived for 1/eta(T,p), the inverse of shear viscosity. The Stokes-Einstein relation for liquids declares that D and T/eta should have a fixed ratio as T and p vary, but experiment reveals substantial violations for deeply supercooled liquids. Upon analyzing the self-diffusion and viscous flow processes in terms of configuration space inherent structures and kinetic transitions between their basins, one possible mechanism for this violation emerges. This stems from the fact that interbasin transitions become increasingly Markovian as T declines, and though self-diffusion is possible in a purely Markovian regime, shear viscosity in the present formulation intrinsically relies on successive correlated transitions.