  We introduce a probabilistic representation of the derivative of the
semigroup associated to a multidimensional killed diffusion process defined on
the half-space. The semigroup derivative is expressed as a functional of a
process that is normally reflected when it hits the hyperplane. The
representation of the derivative also involves a matrix-valued process which
replaces the Jacobian of the underlying process that appears in the traditional
pathwise derivative of a classical diffusion. The components of this
matrix-valued process become zero except for those on the first row every time
the reflected process touches the boundary. The results in this paper extend
those in recent work of the authors, where the one-dimensional case was
studied.
