Wiener path intersections and local time

Abstract

AbstractWe study intersection properties of Wiener processes in the plane. For each positive integer k we show that k independent Wiener processes intersect almost surely in a set of Hausdorff dimension two, and that the set of points a single process visits at least k distinct times also has dimension two. We construct a functional on configurations of k independent Wiener processes that measures the extent to which the trajectories of the k processes intersect. We prove certain Lp estimates for this functional and show that it is a local time for a certain vector-valued multiparameter stochastic process