On the Hitting Probabilities of Limsup Random Fractals

Abstract

Let [Formula: see text] be a limsup random fractal with indices [Formula: see text] and [Formula: see text] on [Formula: see text]. We determine the hitting probability [Formula: see text] for any analytic set [Formula: see text] with the condition [Formula: see text], where [Formula: see text] denotes the Hausdorff dimension. This extends the correspondence of Khoshnevisan et al.1 by relaxing the condition that the probability [Formula: see text] of choosing each dyadic hyper-cube is homogeneous and [Formula: see text] exists. We also present some counterexamples to show the Hausdorff dimension in condition [Formula: see text] cannot be replaced by the packing dimension.

Large intersection property for limsup sets in metric space

“…We can also use the hitting probability of limsup random fractals to get the estimation on Hausdorff dimension in Corollary 4.1. For instance, let A be a limsup random fratcal in T with γ 1 = γ 2 < 1, δ = 0, and if P n (Q) is independent of Q, Khoshnevisan, Peres and Xiao [17] showed that dim H A = 1−γ 1 a.s. For more related research, see [13,15].…”