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Proceedings of the American Mathematical Society

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On a problem of Dynkin

Author: Yuan-chung Sheu
Journal: Proc. Amer. Math. Soc. 127 (1999), 3721-3728
MSC (1991): Primary 60J60, 35K55; Secondary 60J80, 31C45
Published electronically: May 17, 1999
MathSciNet review: 1616617
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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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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

Keywords: Superdiffusion, graph of superdiffusion, semilinear partial differential equation, $\mathbb{G}$-polarity, $\mathbb{H}$-polarity, Hausdorff dimension, box dimension, restricted Hausdorff dimension
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
Article copyright: © Copyright 1999 American Mathematical Society