Exact Hausdorff measures and fine geometry of collisions of independent Markov processes

Abstract: We study the fine geometry of collision measures associated with independent Markov processes on a common metric measure space. These measures are natural space-time random measures encoding collision times and collision points. Under small-scale Ahlfors regularity and standard short-time heat-kernel estimates, we determine the local dimensions of their temporal and spatial marginals, the Hausdorff dimensions of their supports and of the collision sets themselves, and logarithmic limsup laws for their local masses at typical collisions and at exceptional thick collisions. Through this analysis, we identify exact Hausdorff measure functions for the collision-time sets and, in a natural regime, for the collision-point sets. Moreover, we prove that the corresponding Hausdorff measures are comparable, uniformly over all Borel subsets, to the temporal and spatial marginals of the collision measures. This provides a broad solution to the Hausdorff-measure part of an open question posed by Xiao more than two decades ago. Our framework covers symmetric stable processes, canonical diffusions on affine nested fractals such as the Sierpiński gasket, and stable-like jump processes on $d$-sets.