  For a class of stochastic differential equations with reflection for which a
certain ${\mathbb{L}}^p$ continuity condition holds with $p>1$, it is shown
that any weak solution that is a strong Markov process can be decomposed into
the sum of a local martingale and a continuous, adapted process of zero
$p$-variation. When $p=2$, this implies that the reflected diffusion is a
Dirichlet process. Two examples are provided to motivate such a
characterization. The first example is a class of multidimensional reflected
diffusions in polyhedral conical domains that arise as approximations of
certain stochastic networks, and the second example is a family of
two-dimensional reflected diffusions in curved domains. In both cases, the
reflected diffusions are shown to be Dirichlet processes, but not
semimartingales.
