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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On a problem of Dynkin
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by Yuan-chung Sheu PDF
Proc. Amer. Math. Soc. 127 (1999), 3721-3728 Request permission

Abstract:

Consider an $(L,\alpha )$-superdiffusion $X$ on ${\mathbb {R}}^{d}$, where $L$ is an uniformly elliptic differential operator in ${\mathbb {R}}^{d}$, and $1<\alpha \le 2$. The $\mathbb {G}$-polar sets for $X$ are subsets of $\mathbb {R}\times {\mathbb {R}}^{d}$ which have no intersection with the graph $\mathbb {G}$ of $X$, and they are related to the removable singularities for a corresponding nonlinear parabolic partial differential equation. Dynkin characterized the $\mathbb {G}$-polarity of a general analytic set $A\subset \mathbb {R}\times {\mathbb {R}}^{d}$ in term of the Bessel capacity of $A$, and Sheu in term of the restricted Hausdorff dimension. In this paper we study in particular the $\mathbb {G}$-polarity of sets of the form $E\times F$, where $E$ and $F$ are two Borel subsets of $\mathbb {R}$ and ${\mathbb {R}}^{d}$ respectively. We establish a relationship between the restricted Hausdorff dimension of $E\times F$ and the usual Hausdorff dimensions of $E$ and $F$. As an application, we obtain a criterion for $\mathbb {G}$-polarity of $E\times F$ in terms of the Hausdorff dimensions of $E$ and $F$, which also gives an answer to a problem proposed by Dynkin in the 1991 Wald Memorial Lectures.
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Additional Information
  • Yuan-chung Sheu
  • Affiliation: Department of Applied Mathematics, National Chiao-Tung University, Hsinchu, Taiwan
  • Address at time of publication: Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, California 94720-5070
  • Email: ycsheu@nctu.math.edu.tw
  • Received by editor(s): December 1, 1997
  • Received by editor(s) in revised form: February 23, 1998
  • Published electronically: May 17, 1999
  • Communicated by: Stanley Sawyer
  • © Copyright 1999 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 127 (1999), 3721-3728
  • MSC (1991): Primary 60J60, 35K55; Secondary 60J80, 31C45
  • DOI: https://doi.org/10.1090/S0002-9939-99-04981-3
  • MathSciNet review: 1616617