  We present a Brownian dynamics theory with full hydrodynamics (Stokesian
dynamics) for a Gaussian polymer chain embedded in a liquid membrane which is
surrounded by bulk solvent and walls. The mobility tensors are derived in
Fourier space for the two geometries, namely, a free membrane embedded in a
bulk fluid, and a membrane sandwiched by the two walls. Within the preaveraging
approximation, a new expression for the diffusion coefficient of the polymer is
obtained for the free membrane geometry. We also carry out a Rouse normal mode
analysis to obtain the relaxation time and the dynamical structure factor. For
large polymer size, both quantities show Zimm-like behavior in the free
membrane case, whereas they are Rouse-like for the sandwiched membrane
geometry. We use the scaling argument to discuss the effect of excluded volume
interactions on the polymer relaxation time.
