On the existence of self-intersections for quasi-every Brownian path in space
\textit{M. Fukushima} [J. Math. Soc. Japan 36, No.1, 161-176 (1984; Zbl 0522.60081)] defined the notion of quasi-everywhere (q.e.) for standard Brownian motion. The main purpose of the present paper is to show that much of the theory due to \textit{J. Rosen} [Commun. Math. Phys. 88, No.3, 327-338 (1983; Zbl 0534.60070)] on the existence and properties of the self-intersection of almost every Brownian path carries over to results for quasi-every Brownian path. In \(R^ 3\) it is not only shown that quasi-every Brownian path intersects itself in any time interval, but also that the set of double points has Hausdorff dimension 1, thereby strengthening the a.e. result due to \textit{B. Fristedt} [Z. Wahrscheinlichkeitstheorie Verw. Gebiete 9, 62-64 (1967; Zbl 0314.60030)]. In order to show that the self-intersection local time on a bounded time set is q.e. positive, the author requires an upper estimate of the small tail of the distribution of the intersection local time of two independent Brownian motions starting at the same point. The reviewer would like to know the asymptotic behavior of this small tail.
scientific article; zbMATH DE number 4176158