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arXiv · 2112.07135

On the hitting probabilities of limsup random fractals

Zhang-nan Hu·Wen-Chiao Cheng·Bing Li
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Abstract

Let $A$ be a limsup random fractal with indices $\gamma_1, ~\gamma_2 ~$and $\delta$ on $[0,1]^d$. We determine the hitting probability $\mathbb{P}(A\cap G)$ for any analytic set $G$ with the condition $(\star)$$\colon$ $\dim_{\rm H}(G)>\gamma_2+\delta$, where $\dim_{\rm H}$ denotes the Hausdorff dimension. This extends the correspondence of Khoshnevisan, Peres and Xiao [10] by relaxing the condition that the probability $P_n$ of choosing each dyadic hyper-cube is homogeneous and $\lim\limits_{n\to\infty}\frac{\log_2P_n}{n}$ exists. We also present some counterexamples to show the Hausdorff dimension in condition $(\star)$ can not be replaced by the packing dimension.

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Zhang-nan Hu, Wen-Chiao Cheng, Bing Li. 2021-12-14. On the hitting probabilities of limsup random fractals. https://doi.org/10.1142/s0218348x22500554

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