Summary
The manifold metric between two points in a planar domain is the minimum of the lengths of piecewiseC 1 curves in the domain connecting these two points. We define a bounded simply connected planar region to be a pseudo Jordan domain if its boundary under the manifold metric is topologically homeomorphic to the unit circle. It is shown that reflecting Brownian motionX on a pseudo Jordan domain can be constructed starting at all points except those in a boundary subset of capacity zero.X has the expected Skorokhod decomposition under a condition which is satisfied when ∂G has finite 1-dimensional lower Minkowski content.
References
Ahlfors, L.V.: Complex analysis. 3rd edn. New York: MacGraw-Hill 1979
Chen, Z.Q.: On reflecting diffusion processes and Skorokhod decompositions. Probab. Theory Relat. Fields (to appear)
De la Vallée Poussin, C.J.: Application de l'intégrale de Lebesgue au problème de la représentation d'une aire simlement connexe sur un cercle. Ann. Soc. Sci. Br., Sér. A, Sci. Math.50, 23–34 (1930)
Federer, H.: Geometric measure theory. Berlin Heidelberg New York: Springer 1969
Fukushima, M.: A construction of reflecting barrier Brownian motions for bounded domains. Osaka J. Math.4, 183–215 (1967)
Fukushima, M.: Regular representations of Dirichlet spaces. Trans. Am. Math. Soc.155, 455–473 (1971)
Fukushima, M.: Dirichlet forms and symmetric markov processes. Amsterdam: North-Holland 1980
Jones, P.W.: Quasiconformal mappings and extendibility of functions in Sobolev spaces. Acta Math.147, (1–2) 71–88 (1981)
Royden, H.L.: Real analysis. 3rd edn. New York: Macmillan 1988
Rudin, W.: Real and complex analysis. 2nd edn. New York: McGraw-Hill 1974
Silverstein, M.L.: Symmetric Markov processes. (Lect. Notes Math., vol. 426) Berlin Heidelberg New York: Springer 1974
Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton, N.J.: Princeton University Press 1970
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Chen, ZQ. Pseudo Jordan domains and reflecting Brownian motions. Probab. Th. Rel. Fields 94, 271–280 (1992). https://doi.org/10.1007/BF01192446
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DOI: https://doi.org/10.1007/BF01192446
Mathematics Subject Classification
- P 60J65
- S 31C25