Slow Points in the Support of Historical Brownian Motion

Abstract

A slow point from the left for Brownian motion is a time during a given interval for which the oscillations of the path immediately to the left of this time are smaller than the typical ones, that is, those given by the local LIL. These slow points occur at random times during a given interval. For historical super-Brownian motion, the support at a fixed time contains an infinite collection of paths. This paper makes use of a branching process description of the support to investigate the slowness of these paths at the fixed time. The upper function found is the same as that found for slow points in the Brownian motion case.