ISSN:
1572-929X
Keywords:
Primary 46E35
;
60J65
;
Secondary 31C25
;
60G17
;
60J55
;
Reflecting Brownian motion
;
Dirichlet space
;
Sobolev space
;
boundary local time
;
Skorokhod decomposition
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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 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 $$\overline {D\backslash E} $$ 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, W 1,2(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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00275474
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