  This article presents a review of some old and new results on the long time
behavior of reflected diffusions. First, we present a summary of prior results
on construction, ergodicity and geometric ergodicity of reflected diffusions in
the positive orthant $\mathbb{R}^d_+$, $d \in \mathbb{N}$. The geometric
ergodicity results, although very general, usually give implicit convergence
rates due to abstract couplings and Lyapunov functions used in obtaining them.
This leads us to some recent results on an important subclass of reflected
Brownian motions (RBM) (constant drift and diffusion coefficients and oblique
reflection at boundaries), known as the Harrison-Reiman class, where explicit
rates of convergence are obtained as functions of the system parameters and
underlying dimension. In addition, sufficient conditions on system parameters
of the RBM are provided under which local convergence to stationarity holds at
a `dimension-free' rate, that is, for any fixed $k \in \mathbb{N}$, the rate of
convergence of the $k$-marginal to equilibrium does not depend on the dimension
of the whole system. Finally, we study the long time behavior of infinite
dimensional rank-based diffusions, including the well-studied infinite Atlas
model. The gaps between the ordered particles evolve as infinite dimensional
RBM and this gap process has uncountably many explicit product form stationary
distributions. Sufficient conditions for initial configurations to lie in the
weak domain of attraction of the various stationary distributions are provided.
Finally, it is shown that, under conditions, all of these explicit stationary
distributions are extremal (equivalently, ergodic) and, in some sense, the only
product form invariant probability distributions. Proof techniques involve a
pathwise analysis of RBM using explicit synchronous and mirror couplings and
constructing Lyapunov functions.
