Multiparameter multifractional Brownian motion: local nondeterminism and joint continuity of the local times
type: Item
instance of: scholarly article
title: Multiparameter multifractional Brownian motion: local nondeterminism and joint continuity of the local times
zbMATH Open document ID: 1268.60048
DOI: 10.1214/10-AIHP408
author: Antoine Ayache
author: Narn-Rueih Shieh
author: Y. Xiao
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 2013-03-26T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.aihp/1317906500
review text: 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) admi…
reviewed by: Nijolė Kalinauskaitė
Mathematics Subject Classification ID: 60G22
Mathematics Subject Classification ID: 60G17
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 6148897
zbMATH Keywords: multifractional Brownian motion
zbMATH Keywords: local nondeterminism
zbMATH Keywords: local times
zbMATH Keywords: joint continuity
MaRDI profile type: Publication
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OpenAlex ID: W2092323756
description: scientific article; zbMATH DE number 6148897