TY - JOUR

T1 - A Hausdorff measure classification of polar lateral boundary sets for superdiffusions

AU - Sheu, Yuan-Chung

PY - 2000/5/1

Y1 - 2000/5/1

N2 - 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).

AB - 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).

UR - http://www.scopus.com/inward/record.url?scp=33746848579&partnerID=8YFLogxK

U2 - 10.1017/S0305004199004284

DO - 10.1017/S0305004199004284

M3 - Article

AN - SCOPUS:33746848579

VL - 128

SP - 549

EP - 560

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

SN - 0305-0041

IS - 3

ER -