$\R^d$中分支随机游动的重分形分析和Erdös-Rényi大数定律
Multifractal analysis and Erdös-Rényi laws of large numbers for branching random walks in $\R^d$
摘要：
我们重新考虑$\R^d$值分支随机游动平均值的多重分形分析，考虑标准水平集的完全Hausdorff维数的子集，在每个无限分支上，Erdös-Rényi大数定律的量化版本成立。假设行走的增量的指数矩是有限的，我们确实可以同时控制这样的集合，当水平属于可能水平的紧致凸域$I$的内部时，即当它们与所谓的吉布斯测度相关联时，以及当它们属于由与这些吉布斯测度的“临界”版本相关联的水平组成的$\partial I$的子集$(\partial{I})_{\mathrm{crit}}$时。事实证明，给定这两种类型之一的水平，相关的Erdös-Rényi LLN取决于赋予底层Galton-Watson树边界的度量。为了将我们的控制扩展到所有的边界点的情况下，$\partial I\neq (\partial{I})_{\mathrm{crit}}$，我们稍微加强我们的假设上的增量的分布表现出自然分解的$\partial I\setminus (\partial{I})_{\mathrm{crit}}$至多可数许多凸集$J$的仿射维$\le d-1$在每个我们可以基本上减少研究的内部和临界点相关的一些$\R^{\dim J}$值分支随机行走。
英文摘要：
We revisit the multifractal analysis of $\R^d$-valued branching random walks averages by considering subsets of full Hausdorff dimension of the standard level sets, over each infinite branch of which a quantified version of the Erdös-Rényi law of large numbers holds. Assuming that the exponential moments of the increments of the walks are finite, we can indeed control simultaneously such sets when the levels belong to the interior of the compact convex domain $I$ of possible levels, i.e. when they are associated to so-called Gibbs measures, as well as when they belong to the subset $(\partial{I})_{\mathrm{crit}}$ of $\partial I$ made of levels associated to ``critical'' versions of these Gibbs measures. It turns out that given such a level of one of these two types, the associated Erdös-Rényi LLN depends on the metric with which is endowed the boundary of the underlying Galton-Watson tree. To extend our control to all the boundary points in cases where $\partial I\neq (\partial{I})_{\mathrm{crit}}$, we slightly strengthen our assumption on the distribution of the increments to exhibit a natural decomposition of $\partial I\setminus (\partial{I})_{\mathrm{crit}}$ into at most countably many convex sets $J$ of affine dimension $\le d-1$ over each of which we can essentially reduce the study to that of interior and critical points associated to some $\R^{\dim J}$-valued branching random~walk.
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