The Annals of Probability

Hausdorff Measure of Trajectories of Multiparameter Fractional Brownian Motion

Michel Talagrand

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Abstract

Consider $0 < \alpha < 1$ and the Gaussian process $Y(t)$ on $\mathbb{R}^N$ with covariance $E(Y(t)Y(s)) = |t|^{2\alpha} + |s|^{2\alpha} - |t - s|^{2\alpha}$, where $|t|$ is the Euclidean norm of $t$. Consider independent copies $X^1,\ldots,X^d$ of $Y$ and the process $X(t) = (X^1(t),\ldots,X^d(t))$ valued in $\mathbb{R}^d$. In the transient case $(N < \alpha d)$ we show that a.s. for each compact set $L$ of $\mathbb{R}^N$ with nonempty interior, we have $0 < \mu_\varphi(X(L)) < \infty$, where $\mu_\varphi$ denotes the Hausdorff measure associated with the function $\varphi(\varepsilon) = \varepsilon^{N/\alpha} \log \log(1/\varepsilon)$. This result extends work of A. Goldman in the case $\alpha = 1/2$; the proofs are considerably simpler.

Article information

Source
Ann. Probab., Volume 23, Number 2 (1995), 767-775.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176988288

Digital Object Identifier
doi:10.1214/aop/1176988288

Mathematical Reviews number (MathSciNet)
MR1334170

Zentralblatt MATH identifier
0830.60034

JSTOR
links.jstor.org

Subjects
Primary: 60G15: Gaussian processes
Secondary: 60G17: Sample path properties 26B15: Integration: length, area, volume [See also 28A75, 51M25]

Keywords
Haussdorff dimension Brownian motion

Citation

Talagrand, Michel. Hausdorff Measure of Trajectories of Multiparameter Fractional Brownian Motion. Ann. Probab. 23 (1995), no. 2, 767--775. doi:10.1214/aop/1176988288. https://projecteuclid.org/euclid.aop/1176988288


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