  We discuss the hydrodynamic collective effects due to active protein
molecules that are immersed in lipid bilayer membranes and modeled as
stochastic force dipoles. We specifically take into account the presence of the
bulk solvent which surrounds the two-dimensional fluid membrane. Two membrane
geometries are considered: the free membrane case and the confined membrane
case. Using the generalized membrane mobility tensors, we estimate the active
diffusion coefficient and the drift velocity as a function of the size of a
diffusing object. The hydrodynamic screening lengths distinguish the two
asymptotic regimes of these quantities. Furthermore, the competition between
the thermal and non-thermal contributions in the total diffusion coefficient is
characterized by two length scales corresponding to the two membrane
geometries. These characteristic lengths describe the crossover between
different asymptotic behaviors when they are larger than the hydrodynamic
screening lengths.
