Reflecting Brownian motions and a deletion result for Sobolev spaces of order (1, 2)

Let D be an open set in ℝ^d and E be a relatively closed subset of D having zero Lebesgue measure. A necessary and sufficient integral condition is given for the Sobolev spaces W^1,2(D) and W^1,2(D \ E) to be the same. The latter is equivalent to (normally) reflecting Brownian motion (RBM) on D \ E being indistinguishable (in distribution) from RBM on D̄. This integral condition is satisfied, for example, when E has zero (d - 1)-dimensional Hausdorff measure. Therefore it is possible to delete from D a relatively closed subset E having positive capacity but nevertheless the RBM on D \ E is indistinguishable from the RBM on D̄, or equivalently, W^1,2(D \ E) = W^1,2(D). An example of such kind is: D = ℝ² and E is the Cantor set. In the proof of above mentioned results, a detailed study of RBMs on general open sets is given. In particular, a semimartingale decomposition and approximation result previously proved in [3] for RBMs on bounded open sets is extended to the case of unbounded open sets.