The packing measure of the trajectories of multiparameter fractional Brownian motion

Abstract

Let $X\,{=}\,\{X(t), \ t \in {\R^N}\}$ be a multiparameter fractional Brownian motion of index $\alpha$ ($0< \alpha < 1$) in $\R^d.$ We prove that if $N < \alpha d\ $, then there exist positive finite constants $K_1$ and $K_2 $ such that with probability 1, $$ K_1 \le \hbox{$\varphi$-$p(X([0,1]^N))$} \,{\le} \hbox{ $\varphi$-$p({\rm Gr}X([0,1]^N))$} \,{\le}\, K_2$$ where $\varphi(s) = s^{N/\alpha}/(\log \log1/s)^{N/(2 \alpha)}$, $\varphi$-$p(E)$ is the $\varphi$-packing measure of $E$, $X([0, 1]^N)$ is the image and ${\rm Gr}X([0, 1]^N) \,{=}\, \{(t, X(t)); \ t \in [0, 1]^N\}$ is the graph of $X$, respectively. We also establish liminf and limsup type laws of the iterated logarithm for the sojourn measure of $X$.