独立马尔可夫过程碰撞的精确豪斯多夫测度与精细几何
Exact Hausdorff measures and fine geometry of collisions of independent Markov processes
AI总结：
该研究针对公共度量测度空间上独立马尔可夫过程的碰撞测度，确定了相关豪斯多夫维数、测度函数等，解决了Xiao的公开问题，框架覆盖多种过程。
AI中文摘要：
我们研究公共度量测度空间上独立马尔可夫过程相关碰撞测度的精细几何，这类测度是编码碰撞时间与碰撞点的自然时空随机测度。在小尺度阿赫夫斯正则性与标准短时热核估计条件下，我们确定了其时间边缘与空间边缘的局部维数、支撑集及碰撞集本身的豪斯多夫维数，还得到了典型碰撞与异常厚碰撞处局部质量的对数上极限律。通过该分析，我们确定了碰撞时间集的精确豪斯多夫测度函数，且在自然区域内确定了碰撞点集的精确豪斯多夫测度函数；此外，我们证明了对应豪斯多夫测度在所有博雷尔子集上，与碰撞测度的时间边缘、空间边缘是可比的。这为二十多年前Xiao提出的一个公开问题的豪斯多夫测度部分提供了广泛解决方案。我们的框架涵盖对称稳定过程、仿射嵌套分形（如谢尔宾斯基垫片）上的典型扩散过程，以及d-集上类稳定跳跃过程。
英文摘要：
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.
