Packing dimension profiles and Lévy processes (Q2918786)

For a \(\mathbb{R}^d\)-valued Lévy process \(X\) and a Borel subset \(F\subset\mathbb{R}_+\), let \(k_\varepsilon(t):= \mathbb{P}\) \((X(t)\in B(0,\varepsilon))\), NEWLINE\[NEWLINE\overline{\text{Dim}}_k:= \limsup_{\varepsilon\searrow 0}{1\over\log\varepsilon} \log\{\text{inf}\iint k_\varepsilon(|s- t|)\,\nu(ds)\,\nu(dt)\mid \nu(F)= 1,\,\nu(\mathbb{R}\setminus F)= 0\},NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\text{Dim}_k F:= \text{inf}\Biggl\{\sup_{n\in\mathbb{N}}\,\overline{\text{Dim}}_kF_n\mid \bigcup_{n\in\mathbb{N}} F_n= F\Biggr\}.NEWLINE\]NEWLINE Then the nice main result is that almost surely, the upper Minkowski dimension of \(X(F)\) equals \(\overline{\text{Dim}}_kF\) and the packing dimension of \(X(F)\) equals \(\text{Dim}_kF\).NEWLINENEWLINE Corollaries are derived, which yield alternative expansions for the various types of dimensions involved (and some other ones, after \textit{K. J. Falconer} and \textit{J. D. Howroyd} [Math. Proc. Camb. Philos. Soc. 121, No. 2, 269--286 (1997; Zbl 0881.28002) and \textit{J. D. Howroyd} [ibid. 130, No. 1, 135--160 (2001; Zbl 1007.28004)]).