  We determine the processes obtained from a large class of reflected Brownian
motions (RBMs) in the nonnegative orthant by means of time reversal. The class
of RBMs we deal with includes, but is not limited to, RBMs in the so-called
Harrison-Reiman class [4] having diagonal covariance matrices. For such RBMs
our main result resolves the long-standing open problem of determining the time
reversal of RBMs beyond the skew-symmetric case treated by R.J. Williams in
[16]. In general, the time-reversed process itself is no longer a RBM, but its
distribution is absolutely continuous with respect to a certain auxiliary RBM.
In the course of the proof we introduce a novel discrete approximation scheme
for the class of RBMs described above, and use it to determine the semigroups
dual to the semigroups of such RBMs.
