A representation of local time for Lipschitz surfaces

Abstract

Suppose that D [is an element of the set of Real numbers to the power of n], n [is greater than or equal to] 2, is a Lipschitz domain and let N[subscript]t(r) be the number of excursions of Brownian motion inside D with diameter greater than r which started before time t. Then rN[subscript]t(r) converges as r --> 0 to a constant multiple of local time on [partial derivative]D, a.s. and in L[to the power of]p for all p < infinity. The limit need not exist or may be trivial (0 or 1) in Hölder domains, non-tangentially accessible domains and domains whose boundaries have finite surface area.