Summary.
We consider a \(d\)-dimensional Euclidean domain \(D\) whose boundary is Lipschitz continuous but admits locally finite number of outward or inward Hölder cusp points. Using a method of Stampacchia and Moser for PDE, we first construct a conservative diffusion process on the Euclidean closure of \(D\) possessing a strong Feller resolvent and associated with a second order uniformly elliptic differential operator of divergence form with measurable coefficients \(a_{ij}\). The sample path of the constructed diffusion can be uniquely decomposed as a sum of a martingale additive functional and an additive functional locally of zero energy. The second additive functional will be proved to be of bounded variation with a Skorohod type expression whenever \(a_{ij}\) is weakly differentiable and the Hölder exponent at each outward cusp boundary point is greater than \(1/2\) regardless the dimension \(d\).
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Received: 4 October 1995
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Fukushima, M., Tomisaki, M. Construction and decomposition of reflecting diffusions on Lipschitz domains with Hölder cusps. Probab Theory Relat Fields 106, 521–557 (1996). https://doi.org/10.1007/s004400050074
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DOI: https://doi.org/10.1007/s004400050074