On the multiple time set of Brownian motions (Q1333947)

Let \(B^ d_ t\), \(t \geq 0\), be Brownian motion in \(R^ d\) starting at the origin and \(S^ 2_ d\) its self-intersection time set, i.e. \(S^ 2_ d = \{(t_ 1,t_ 2) \in R^ 2_ + : 0 \leq t_ 1 < t_ 2 < \infty\) and \(B^ d_{t_ 1} = B^ d_{t_ 2}\}\). For \(d=3\) it is proved that the Hausdorff measure function for \(S^ 2_ 3\) is \(\varphi_ 3^{(2)}(t) = t^{1/2} (\log \log t)^{3/2}\). This establishes for the particular case \(d=3\) a conjecture put forward by \textit{J. Rosen} [Commun. Math. Phys. 88, 327-338 (1983; Zbl 0534.60070)].