Multiparameter multifractional Brownian motion: local nondeterminism and joint continuity of the local times
The aim of the paper is to provide a method for establishing the local nondeterminism property for multifractional Brownian motions (mfBm) of harmonisable-type. The method originates from wavelet analysis and differs from the existing methods in the literature. A harmonisable-type isotropic multiparameter mfBm with values in \(\mathbb{R}\) is defined by \[ X(t):= \int_{\mathbb{R}^N} \frac{e^{i(t,\xi)-1}}{|\xi|^{H(t)+\frac{N}{2}}} d\hat{W}(\xi) \quad \text{for every }t \in \mathbb{R}^N, \] where \(H(\cdot)\) is a functional parameter with values in a fixed interval \([a,b] \subset (0,1)\), satisfying a uniform Hölder condition of order \(\beta = \beta(I) \in (b,1]\) on any compact cube \(I \subset \mathbb{R}^N\) and \(d\hat{W}\) is the Fourier transform of the real-valued white noise \(dW\). The Gaussian field \(X=\{X(t), t \in \mathbb{R}^N\}\) can be represented as a certain random wavelet-type series. After the Introduction, the local nondeterminism of the multifractional Brownian motion \({X}\) is studied in Section 2. In order to prove that \(X=\{X(t), t \in \mathbb{R}^N\}\) is locally nondeterministic on closed bounded rectangles \(I \subset \mathbb{R}^N_+\), it is sufficient to show that, for any integer \(m \geq 1\), there exists a constant \(c^{(m)}>0\) such that, for all \(t^0,t^1,\dotsc, t^m \in I\) and all real numbers \(\alpha_1,\dotsc,\alpha_m\), \[ \operatorname{E} \left|X(t^0) - \sum^m_{n=1} \alpha_n X(t^n)\right|^2 \geq c^{(m)} \min\{|t^n-t^0|^{2H(t^0)}, 1\leq n \leq m\} \] holds, which is the contents of Theorem 2.1 and which is proved using the wavelet method. In Section 3, Theorem 2.1 is applied to prove the joint continuity of local times of a Gaussian random field in \(\mathbb{R}^N\) defined by \[ X(t) = (X_1(t), \dotsc, X_d(t)), \] where \(X_i\;(l \leq i \leq d)\) are independent copies of the real-valued multifractional Brownian motion with index function \(H({t})\) as defined above. Theorem 3.1 states that such a random motion \({X}\) (i) admits an \(L^2(\mathbb{R}^d)\)-integrable local time \(L(\cdot,I)\) almost surely; and (ii) has a jointly continuous local time on \({I}\) if \(I=[\varepsilon,1]^N\) and \(\bar{H} = \max_{t \in I} X(t)\), and \(N>\bar{H}d\). Theorem 3.5 deals with local and uniform Hölder conditions for an \((N,d)\)-multifractional Brownian motion under same conditions as in Theorem 3.1. The section ends with two applications of Theorem 3.5. Theorem 3.6 deals with mfBm with values in \(\mathbb{R}\) and, in particular, it states that the sample paths are almost surely nowhere differentiable in \({I}\). It states that, for any \(\delta>0\), \[ \lim_{r \rightarrow 0} \inf \sup_{t \in \mathbb{B} (\tau,r)} \frac{X(t) - X(\varepsilon)}{r^{H(\tau)+\delta}} \geq c_1 \quad\text{a.s.} \] and \[ \lim_{r \rightarrow 0} \inf \inf_{\tau \in I} \sup_{t \in \mathbb{B} (\tau,r)} \frac{X(t) - X(\varepsilon)}{r^{\bar{H}+\delta}} > c_2 \quad \text{a.s., where } \bar{H}=\max_{t \in I} H(t). \] The second application of Theorem 3.5 is Theorem 3.8 on the Hausdorff dimension of the level sets of a \((N,d)\)-multifractional Brownian motions.
scientific article; zbMATH DE number 6148897
scientific article