On the hitting probabilities of limsup random fractals

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 )$$:$ ${\dim }_{\mathrm{H}}(G)>{\gamma }_2+\delta$, where ${\dim }_{\mathrm{H}}$ 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 $\underset{n\to \infty }{\lim }\frac{{\log }_2{P}_n}{n}$ exists. We also present some counterexamples to show the Hausdorff dimension in condition $(\star )$ can not be replaced by the packing dimension.

12 pages