  Active motions of a biological membrane can be induced by non-thermal
fluctuations that occur in the outer environment of the membrane. We discuss
the dynamics of a membrane interacting hydrodynamically with an active wall
that exerts random velocities on the ambient fluid. Solving the hydrodynamic
equations of a bound membrane, we first derive a dynamic equation for the
membrane fluctuation amplitude in the presence of different types of walls.
Membrane two-point correlation functions are calculated for three different
cases; (i) a static wall, (ii) an active wall, and (iii) an active wall with an
intrinsic time scale. We focus on the mean squared displacement (MSD) of a
tagged membrane describing the Brownian motion of a membrane segment. For the
static wall case, there are two asymptotic regimes of MSD ($\sim t^{2/3}$ and
$\sim t^{1/3}$) when the hydrodynamic decay rate changes monotonically. In the
case of an active wall, the MSD grows linearly in time ($\sim t$) in the early
stage, which is unusual for a membrane segment. This linear-growth region of
the MSD is further extended when the active wall has a finite intrinsic time
scale.
