On a problem of Dynkin

Abstract

Consider an (L, alpha)-superdiffusion X on R-d, where L is an uniformly elliptic differential operator in R-d, and 1 < alpha less than or equal to 2. The G-polar sets for X are subsets of R x R-d 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 subset of R x 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 G-polarity of sets of the form E x F, where E and F are two Borel subsets of R and R-d respectively. We establish a relationship between the restricted Hausdorff dimension of E x F and the usual Hausdorff dimensions of E and F. As an application, we obtain a criterion for G-polarity of E x 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.