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

Abstract

Let D be an open set in ℝd 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 W1,2 (D) and W1,2(D\E) to be the same. The latter is equivalent to (normally) reflecting Brownian motion (RBM) on \(\overline {D\backslash E} \) being indistinguishable (in distribution) from RBM on \(\bar D\). 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 \(\overline {D\backslash E} \) is indistinguishable from the RBM on \(\bar D\), or equivalently, W1,2(D\E)=W1,2(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.