On a problem of Dynkin

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.