On Brownian Excursions in Lipschitz Domains. Part I. Local Path Properties

Abstract

A necessary and sufficient condition is given for a Brownian excursion law in a Lipschitz domain to share the local path properties with an excursion law in a halfspace. This condition is satisfied for all boundary points of every C1't-domain, ct > 0. There exists a Cl-domain such that the condition is satisfied almost nowhere on the boundary. A probabilistic interpretation and applications to minimal thinness and boundary behavior of Green functions are given. 0. Introduction. We will show that the local path properties of Brownian excursions in Cl,--domains, a > 0, are the same as the local path properties of Brownian excursions in a half-space (Theorem 3.1). This need not be the case if the domain is of class C' (Proposition 3.1). The method of proof is based on the exit system theory of Maisonneuve [12]. We will study Brownian excursion laws, i.e., one of the ingredients of an exit system. We will give a necessary and sufficient condition for an excursion law in a Lipschitz domain to have the same local path properties as an excursion law in a half-space (Theorems 2.1 and 2.2). This last result can be applied to obtain new criteria for minimal thinness (Theorem 4.1) and existence of a nondegenerate normal derivative for the Green function (Theorem 4.2). Local path properties of 1-dimensional Brownian excursions have been known for some time. They are the same as the local path properties of the 3-dimensional Bessel process (see ?2.10 of Ito and McKean [11] or ?II.67 of Williams [21]). Dvoretsky and Erd6s [8] and Shiga and Watanabe [18] have given Kolmogorov-type tests for Bessel processes. Local path properties of excursions of multidimensional Brownian motion have been studied by Shimura [19] (excursions of 2-dimensional Brownian motion from a line) and Burdzy [1] (n-dimensional case, n > 2). Our theorem about minimal thinness generalizes results of Essen and Jackson [9] and Burdzy [3, 4]. A new criterion for the existence of the nondegenerate normal derivative for the Green function extends some results of Widman [20], Rippon [17] and Burdzy [4]. The present article mainly uses ideas from excursion theory (see Maisonneuve [12], Williams [21] and Burdzy [1]) and potential theory (see Doob [7]). Received by the editors July 26, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 60G17, 60J50, 60J65; Secondary 31B25, 60J45.