Fractional Brownian sheets run with nonlinear clocks

Abstract

Let \documentclass[12pt]{minimal}\begin{document}$X=\lbrace X(t),\,t\in {\mathbb R}_+^N\rbrace$\end{document}X={X(t),t∈R+N} be an (N, d)-fractional Brownian sheet run with nonlinear clocks, that is, X(t) = BH(F1(t1), …, FN(tN)) for all \documentclass[12pt]{minimal}\begin{document}$t=(t_1,\ldots ,t_N)\in {\mathbb R}_+^N$\end{document}t=(t1,...,tN)∈R+N, where BH is an (N, d)-fractional Brownian sheet with Hurst indices H = (H1, …, HN) ∈ (0, 1)N, and where \documentclass[12pt]{minimal}\begin{document}$F_i:\, {\mathbb R}_+\rightarrow {\mathbb R}_+$\end{document}Fi:R+→R+ (i = 1, …, N) are N non-negative nondecreasing functions that satisfy bi-Lipschitz conditions. In this paper, we study sample path properties of fractional Brownian sheet run with nonlinear clocks X. We first derive a sharp modulus of continuity for X, and then determine the Hausdorff and packing dimensions of the range X([0, 1]N), the graph GrX([0, 1]N), and the level sets for the X. We also provide necessary and sufficient conditions for the existence of local times of X and investigate the joint continuity of the local times. Finally, we study the intersection behavior of two independent fractional Brownian sheets run with (possibly different) nonlinear clocks.