On a Problem of Dynkin

Consider an (L, α)-superdiffusion X on Rd, where L is an uniformly elliptic differential operator in Rd, and 1 < α ≤ 2. The G-polar sets for X are subsets of R×Rd which have no intersection with the graph G of X, and they are related to the removable singularities for a corresponding nonlinear parabolic partial differential equation. Dynkin characterized the G-polarity of a general analytic set A ⊂ R × Rd 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 G-polarity of sets of the form E × F , where E and F are two Borel subsets of R and Rd respectively. We establish a relationship between the restricted Hausdorff dimension of E × F and the usual Hausdorff dimensions of E and F . As an application, we obtain a criterion for G-polarity of E × 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.