Nonnegativity of solutions to the basic adjoint relationship for some diffusion processes

For a multi-dimensional diffusion process, an important problem is whether the associated basic adjoint relationship (BAR) uniquely characterizes the stationary distribution of the diffusion process. A key step in this characterization is an open problem that any solution to BAR does not change sign. This note describes the open problem precisely in the context of two classes of diffusion processes. They are semimartingale reflecting Brownian motions and piecewise Ornstein–Uhlenbeck processes.