  It has recently been demonstrated that black hole dynamics at large D is dual
to the motion of a probe membrane propagating in the background of a spacetime
that solves Einstein's equations. The equation of motion of this membrane is
determined by the membrane stress tensor. In this paper we `improve' the
membrane stress tensor derived in earlier work to ensure that it defines
consistent probe membrane dynamics even at finite $D$ while reducing to
previous results at large D. Our improved stress tensor is the sum of a Brown
York term and a fluid energy momentum tensor. The fluid has an unusual equation
of state; its pressure is nontrivial but its energy density vanishes. We
demonstrate that all stationary solutions of our membrane equations are
produced by the extremisation of an action functional of the membrane shape.
Our action is an offshell generalization of the membrane's thermodynamical
partition function. We demonstrate that the thermodynamics of static spherical
membranes in flat space and global AdS space exactly reproduces the
thermodynamics of the dual Schwarzschild black holes even at finite D. We study
the long wavelength dynamics of membranes in AdS space, and demonstrate that
the boundary `shadow' of this membrane dynamics is boundary hydrodynamics with
with a definite constitutive relation. We determine the explicit form of shadow
dual boundary stress tensor upto second order in derivatives of the boundary
temperature and velocity, and verify that this stress tensor agrees exactly
with the fluid gravity stress tensor to first order in derivatives, but
deviates from the later at second order and finite D.
