A representation of local time for Lipschitz surfaces (Q1118912)

Suppose that \(D\subset {\mathbb{R}}^ n\), \(n\geq 2\), is a Lipschitz domain and let \(N_ t(r)\) be the number of excursions of Brownian motion inside D with diameter greater than r which started before time t. Then \(rN_ t(r)\) converges as \(r\to 0\) to a constant multiple of local time on \(\partial D\), a.s. and in \(L^ p\) for all \(p<\infty\). The limit need not exist or may be trivial (0 or \(\infty)\) in Hölder domains, non- tangentially accessible domains and domains whose boundaries have finite surface area.