A cluster of sets of exceptional times of linear Brownian motion
This paper considers weak maxima and points of increase of linear Brownian motion. To give an example of weak maxima, let \(V(\alpha ,x)=\text{sign}(x) |x |^\alpha\). Then \(t\) is a weak maximum if \[ \int_s^t V(\alpha_1,B_u-B_t) du \leq 0\;\forall s\in [0,t)\;\text{ and } \int V(\alpha_2, B_u-B_t) du \leq 0\;\forall s\in (t,1]. \] It is shown that the set of weak maxima in this sense has Hausdorff dimension \(1-(\rho_1+\rho_1)/2\) with positive probability, where \(\rho_i\) solves \(\sin \frac{\pi(1-\rho_i)}{2+\alpha_i} =\sin \frac{\pi\rho_i}{2+\alpha_i}.\) Different results of this type are presented, which all generalize a result of \textit{S. Aspadiiarov} and \textit{J-F. Le Gall} [Ann. Probab. 23, No. 4, 1605-1626 (1995; Zbl 0852.60091)] and use similar methods.
scientific article; zbMATH DE number 1747463