For Brownian motion contained in a closed region (specifically a polyhedron), is it true in general that the stationary density on the boundary is continuous with the value on the interior, in a sense similar to "left continuity" in 1D?
I think, but am not sure, that my question is: for a region $A$, is the value of the density $p(x)$ at a point $x$ on the boundary $\partial A$ equal to the $\lim_{s\rightarrow \infty} p(y(s))$ for all $y:\mathbb{R} \rightarrow A$ with $y(s) \rightarrow x$ as $s\rightarrow \infty$?
Harrison and Williams 1987 [https://projecteuclid.org/journals/annals-of-probability/volume-15/issue-1/Multidimensional-Reflected-Brownian-Motions-Having-Exponential-Stationary-Distributions/10.1214/aop/1176992259.pdf] show an exponential closed form that (AFAICT) holds on both the interior and boundary.
Also, (15) of Dai and Harrison 1991 [https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=ce39ac73d4960c287461cf26855b975587941c98] is a single integral equation is written for the density on a closed rectangle by defining quantities differently on the boundary and interior. That suggests that it is continuous.
However, I know from [https://math.stackexchange.com/questions/150097/must-probability-density-be-continuous/150460] and [https://math.stackexchange.com/questions/57317/construction-of-a-borel-set-with-positive-but-not-full-measure-in-each-interval] that the density needn't be continuous anywhere. (How can I provide cleaner links to items on MSE? The preview while I'm editing replaces these links by the item titles, but when I submitted, it put the raw links back ):
I'm particularly looking for a citable source and/or somewhere that I can read more about the invariant measure of these processes, rather than just an answer to this particular question.
