## On a problem of Dynkin

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**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.## References

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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