Let $G \subset \mathbb{R}^k$ be a convex polyhedral cone with vertex
at the origin given as the intersection of half spaces $\{G_i, i =1,\ldots,
N\}$, where $n_i$ and $d_i$ denote the inward normal and direction of
constraint associated with $G_i$, respectively. Stability properties of a class
of diffusion processes, constrained to take values in $G$, are studied under
the assumption that the Skorokhod problem defined by the data $\{(n_i,d_i),i =
1,\ldots,N\}$ is well posed and the Skorokhod map is Lipschitz continuous.
Explicit conditions on the drift coefficient, $b(\cdot)$, of the diffusion
process are given under which the constrained process is positive recurrent and
has a unique invariant measure.Define
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¶ Then the key condition for stability is that there exists $\delta
\in (0,\infty)$ and a bounded subset $A$ of $G$ such that for all $x \in
G\setminus A, b(x) \in \mathcal{C}$ and $\dist (b(x),\partial\mathcal{C}) \ge
\delta$, where $\partial\mathcal{C}$denotes the boundary of $\mathcal{C}$.