Sample Paths Properties of the Set-Indexed Fractional Brownian Motion

Abstract

For \(0 < H \le 1/2\), let \(\mathbf {B}^H = \{ \mathbf {B}^H(t);\; t\in \mathbb R^N_+ \}\) be the Gaussian random field obtained from the set-indexed fractional Brownian motion restricted to the rectangles of \(\mathbb R^N_+\). We prove that \(\mathbf {B}^H\) is tangent to a multiparameter fBm which is isotropic in the \(l^1\)-norm and we determine the Hausdorff dimension of the inverse image of \(\mathbf {B}^H\) and its hitting probabilities. By applying the Lamperti transform and a Fourier analytic method, we show that \(\mathbf {B}^H\) has the property of strong local nondeterminism (SLND) for \(N=2\). By applying SLND, we obtain the exact uniform and local moduli of continuity and Chung’s law of iterated logarithm for \(\mathbf {B}^H = \{ \mathbf {B}^H(t);\; t\in \mathbb R^2_+ \}\). These results show that, away from the axes of \(\mathbb R^2_+\), the local behavior of \(\mathbf {B}^H\) is similar to the ordinary fractional Brownian motion of index H.