  A semi-martingale reflecting Brownian motion is a popular process for
diffusion approximations of queueing models including their networks. In this
paper, we are concerned with the case that it lives on the nonnegative
half-line, but the drift and variance of its Brownian component discontinuously
change at its finitely many states. This reflecting diffusion process naturally
arises from a state-dependent single server queue, studied by the author
(2024). Our main interest is in its stationary distribution, which is important
for application. We define this reflecting diffusion process as the solution of
a stochastic integral equation, and show that it uniquely exists in the weak
sense. This result is also proved in a different way by Atar, Castiel and
Reiman (2022,2023). In this paper, we consider its Harris irreducibility and
stability, that is, positive recurrence, and derive its stationary distribution
under this stability condition. The stationary distribution has a simple
analytic expression, likely extendable to a more general state-dependent SRBM.
Our proofs rely on the generalized Ito formula for a convex function and local
time.
