On the Existence of Self-Intersections for Quasi-Every Brownian Path in Space

Abstract

The set of self-intersections of a Brownian path $b(t)$ taking values in $\mathbb{R}^3$ has Hausdorff dimension 1, for almost every such path, with respect to Wiener measure, a result due to Fristedt. Here we prove that this result (together with the corresponding result for paths in $\mathbb{R}^2$) in fact holds for quasi-every path with respect to the infinite-dimensional Ornstein-Uhlenbeck process, a diffusion process on Wiener space whose stationary measure is Wiener measure. We do this using Rosen's self-intersection local time, first proving that this exists for quasi-every path.