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Construction and decomposition of reflecting diffusions on Lipschitz domains with Hölder cusps
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  • Published: 01 October 2002

Construction and decomposition of reflecting diffusions on Lipschitz domains with Hölder cusps

  • Masatoshi Fukushima1 &
  • Matsuyo Tomisaki2 

Probability Theory and Related Fields volume 106, pages 521–557 (1996)Cite this article

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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Authors and Affiliations

  1. Department of Mathematical Science, Faculty of Engineering Science, Osaka University, Toyonaka, Osaka, Japan (e-mail: fuku@sigmath.es.osaka-u.ac.jp), Osaka, Japan

    Masatoshi Fukushima

  2. Department of Mathematics, Faculty of Education, Yamaguchi University, Yamaguchi, Japan (e-mail: tomisaki@po.yb.cc. yamaguchi-u.ac.jp), Yamaguchi, Japan

    Matsuyo Tomisaki

Authors
  1. Masatoshi Fukushima
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  2. Matsuyo Tomisaki
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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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  • Published: 01 October 2002

  • Issue Date: December 1996

  • DOI: https://doi.org/10.1007/s004400050074

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  • Mathematics Subject Classification (1991): 60J60, 60J55, 35J25, 31C25
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