A Hausdorff measure classification of polar lateral boundary sets for superdiffusions

Abstract

Consider an (L, α)-superdiffusion X, 1 < α [les ] 2, in a smooth cylinder Q = ℝ₊ × D. Where L is a uniformly elliptic operator on ℝ₊ × ℝ^d and D is a bounded smooth domain in ℝ^d. Criteria for determining which (internal) subsets of Q are not hit by the graph [Gscr ] of X were established by Dynkin [5] in terms of Bessel capacity and according to Sheu [14] in terms of restricted Hausdorff dimension (partial results were also obtained by Barlow, Evans and Perkins [3]). While using Poisson capacity on the lateral boundary ∂Q of Q, Kuznetsov [10] recently characterized complete subsets of ∂Q which have no intersection with [Gscr ]. In this work, we examine the relations between Poisson capacity and restricted Hausdorff measure. According to our results, the critical restricted Hausdorff dimension for the lateral [Gscr ]-polarity is d − (3 − α)/(α − 1). (A similar result also holds for the case d = (3 − α)/(α − 1)). This investigation provides a different proof for the critical dimension of the boundary polarity for the range of X (as established earlier by Le Gall [12] for L = Δ, α = 2 and by Dynkin and Kuznetsov [7] for the general case).