DIMENSION OF POLAR SETS FOR BROWNIAN SHEET

摘要

Let W~{W(t);t ∈ RN+} be the d-dimensional N-parameter Brownian Sheet.Sufficient conditions for a compact set F Rd {0} to be a polar set for W are proved.It is also proved that if 2N ≤ d, then for any compact set E R＞N,inf{dimF: F ∈ B(Rd), P{W(E) ∩ F ≠φ} ＞ 0} = d - 2DimE,and if 2N ＞ d, then for any compact set F Rd {0},inf{dimE: E ∈ B(R＞N),P{W(E) ∩ F ≠φ} ＞ 0} = d/2 DimF/22 2 ,where B(Rd) and B(RN＞) denote the Borel σ-algebra in Rd and R＞N respectively, and dimand Dim are Hausdorff dimension and Packing dimension respectively.