Multiple points of fractional Brownian motion and Hausdorff dimension (Q1178402)

Let \(X(t)\) \((t\in R^ N)\) be \(d\)-dimensional fractional Brownian motion of index \(\alpha\) \((0<\alpha<1)\). The Hausdorff dimension of \(k\) multiple points is considered. Let \[ L_ k=\{x\in R^ d: \exists \hbox{ distinct } t_ 1,\ldots,t_ k\in R^ N, \hbox{ such that } X(t_ 1)=X(t_ 2)=...=X(t_ k)=x\}. \] If \(N\leq \alpha d\), \(Nk>(k-1)\alpha d\), then \[ P\{\dim L_ k=Nk/\alpha-(k-1)d\}>0. \] If \(N>\alpha d\), then dim \(L_ k=d\) a.s. For disjoint compact sets \(E_ 1,\ldots,E_ k\) in \(R^ N\backslash\{0\}\), if \(N\leq\alpha d\), we obtain an upper bound for \(\dim(X(E_ 1)\cap\ldots\cap X(E_ k))\) and a lower bound for \(\dim(X(E_ 1)\cap X(E_ 2))\). This proves a conjecture of \textit{Testard} (Thèse, 1987) in the case of \(k=2\) and \(N\leq\alpha d\).