The exact packing measure of Brownian double points
The paper is devoted to the investigations of the packing measure of the doubling points in \(\mathbb{R}^3\). The first main result of the paper states that if \(\xi=\xi_3(2,2)\) is the intersection exponent of two packets of two independent Brownian motions, then a.s. the \(\phi\)-packing measure of a set \(D\) is zero if \[ \int_{0+}^\infty r^{-1-\xi}\phi^\xi(r)~dr<\infty, \] and infinity otherwise. Let \(l_{\vec{t}}(U)\), \(U\subset \mathbb{R}^d\), \(d=2,3\), be the intersection local time of two unions of paths \(\mathcal{B}^{(1)}(t^{(1)},\dots,t^{(M)})\) and \(\mathcal{B}^{(2)}(t^{(M+1)},\dots,t^{(M+N)})\). Here the paths are taken up to the time \(\vec{t}=(t^{(1)},\dots,t^{(M+N)})\). The second main result of the paper is that the probability \(P(l_{\vec{t}} (U)<a)\) is equivalent to \(a^{\frac{\xi}{4-d}}\) for all \(a\in (0,1)\). The proof is relied on the ``branching tree heuristics'' [see also \textit{P. Mörters} and \textit{M. Ortgiese}, Bernoulli 14, No. 1, 277--299 (2008; Zbl 1163.60040)], which describes the strategy how the Brownian path achieves the event \(\{l_{\vec{t}}\}\).
scientific article; zbMATH DE number 5375854
scientific article