A Hausdorff measure classification of polar lateral boundary sets for superdiffusions

Abstract

Consider an (L, alpha)-superdiffusion X, 1 < alpha less than or equal to 2, in a smooth cylinder Q = R+ x D. Where L is a uniformly elliptic operator on R+ x 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 G 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 partial derivative Q of Q, Kuznetsov [10] recently characterized complete subsets of partial derivative Q which have no intersection with G. 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 G-polarity is d - (3 - alpha)/(alpha - 1). (A similar result also holds for the case d = (3 - alpha)/(alpha - 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 aall [12] for L = Delta, alpha = 2 and by Dynkin and Kuznetsov [7] for the general case).