Capacity Estimates, Boundary Crossings and the Ornstein–uhlenbeck Process in Wiener Space

Let T1 denote the first passage time to 1 of a standard Brownian motion. It is well known that as λ → ∞, P{T1 > λ} ∼ cλ−1/2, where c = (2/π). The goal of this note is to establish a capacitarian version of this result. Namely, we will prove the existence of positive and finite constants K1 and K2 such that for all λ > e, K1λ −1/2 ≤ Cap{T1 > λ} ≤ K2λ−1/2 log(λ) · log log(λ), where ‘log’ denotes the natural logarithm, and Cap is capacity on Wiener space. 1Supported, in part, by the Hungarian National Foundation for Scientific Research, Grant No. T 019346 and T 029621 and by the joint French-Hungarian Intergovernmental Grant “Balaton” No. F25/97. 2Supported, in part, by a grant from NSF and by NATO grant No. CRG 972075. 3Supported, in part, by the joint French-Hungarian Intergovernmental Grant “Balaton” No. F25/97 and by NATO grant No. CRG 972075.