On the Explosion of the Local Times Along Lines of Brownian Sheet Étude de l’explosion des temps locaux du drap brownien le long des droites

Abstract: 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 [20] 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 [10]: 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 [14, 18, 20].