Consider an (L, α)-superdiffusion X, 1 < α ≤ 2, in a smooth cylinder Q = ℝ+ × D. Where L is a uniformly elliptic operator on ℝ+ × ℝd and D is a bounded smooth domain in ℝd. Criteria for determining which (internal) subsets of Q are not hit by the graph script G sign of X were established by Dynkin  in terms of Bessel capacity and according to Sheu  in terms of restricted Hausdorff dimension (partial results were also obtained by Barlow, Evans and Perkins ). While using Poisson capacity on the lateral boundary ∂Q of Q, Kuznetsov  recently characterized complete subsets of ∂Q which have no intersection with script G sign. 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 script G sign-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  for L = Δ, α = 2 and by Dynkin and Kuznetsov  for the general case).
|Number of pages||12|
|Journal||Mathematical Proceedings of the Cambridge Philosophical Society|
|State||Published - 1 May 2000|