On the Explosion of the Local Times of Brownian Sheet Along Lines

One can view a 2-parameter Brownian sheet {W (s, t); s, t ≥ 0} as a stream of interacting Brownian motions {W (s, •); s ≥ 0}. Given this viewpoint, we aim to continue the analysis of Walsh (1978) on the local times of the stream W (s, •), near time s = 0. Our main result is a kind of maximal inequality that, in particular, verifies the following conjecture of Khoshnevisan (1995a): as s → 0, the local times of W (s, •) explode almost surely. Two other applications of this maximal inequality are presented, one to a capacity estimate in classical Wiener space, and one to a uniform ratio ergodic theorem in Wiener space. The latter readily implies a quasi-sure ergodic theorem. We also present a sharp Hölder condition for the local times of the mentioned Brownian streams that refines earlier results of (Lacey 1990; Révész 1985; Walsh 1978).