A packing dimension theorem for Gaussian random fields (Q2518957)

Let \(X= \{X_1(t),X_2(t),\dots ,X_d(t)\), \(t\in\mathbb R^N\} \) be a Gaussian random field with values in \(\mathbb R^{d}\) and centered independent identically distributed components. Let the random field \(X_1(t)\), \(t\in\mathbb R^N\) satisfies condition \((C)\). There exist positive constants \(\delta_0\), \(K\geq 1\) and a right-continuous function \(\Phi:[0,\delta )\rightarrow [0,\infty )\) such that \(\Phi (0)=0\) and for all \(t\in\mathbb R^{N}\) and \( h\in\mathbb R^{N}\) with \(\|h\|\leq\delta _{0}\) \(K^{-1}\Phi ^{2}(\|h\|)\leq E[(X_1(t+h)-X_1(t))^2]\leq K\Phi^2(\;h\|)\). Denote the upper index of \(\Phi\) at 0 \[ \alpha^*= \inf \left\{ \beta \geq 0:\lim_{r\downarrow 0} \frac{\Phi (r)}{r^{\beta }}= \infty \right\} \] and lower index \[ \alpha_*=\sup \left\{\beta \geq 0:\lim_{r\downarrow 0} \frac{\Phi (r)}{r^{\beta }}=0\right\} . \] The main result of the paper is the following. Under condition \((C)\) and \( 0<\alpha _{\ast }\leq \alpha ^{\ast }\leq 1\), if \(\Phi \) satisfies either one of the following conditions: For any \(\varepsilon >0\) small enough, there exists a constant \(K\) that for all \(a\in (0,1]\) either \[ \int_0^1 \left(\frac{\Phi(a)}{\Phi(ax)}\right)^dx^{N-1}\,dx\leq Ka^{-\varepsilon}, \] or \[ \int_1^{\frac 1a} \left(\frac{\Phi(a)}{\Phi(ax)}\right)^dx^{N-1}\,dx\leq Ka^{-\varepsilon }, \] then the packing dimension of the range \[ \dim _{p}X([0,1]^{N})=\min \left(d,\frac{N}{\alpha_*}\right) \] a.s. and for the graph \[ \dim _{p} \operatorname{Gr}X([0,1]^N)= \min \left(\frac{N}{\alpha_*},N+(1-\alpha_*)d \right) \quad\text{a.s.} \] The history of the problem and references are discussed in the Introduction and Section 2. At its end some comments on unsolved problems and suggestions are discussed.