A Hausdorff measure classification of polar lateral boundary sets for superdiffusions

Yuan-Chung Sheu*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

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 [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 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 [12] for L = Δ, α = 2 and by Dynkin and Kuznetsov [7] for the general case).

Original languageEnglish
Pages (from-to)549-560
Number of pages12
JournalMathematical Proceedings of the Cambridge Philosophical Society
Volume128
Issue number3
DOIs
StatePublished - 1 May 2000

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