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Images of Gaussian random fields: Salem sets and interior points

Narn-Rueih Shieh;  Yimin Xiao — 2006

Studia Mathematica

Let X = X ( t ) , t ∈ ℝ N be a Gaussian random field in ℝ d with stationary increments. For any Borel set E ⊂ ℝ N , we provide sufficient conditions for the image X(E) to be a Salem set or to have interior points by studying the asymptotic properties of the Fourier transform of the occupation measure of X and the continuity of the local times of X on E, respectively. Our results extend and improve the previous theorems of Pitt [24] and Kahane [12,13] for fractional Brownian motion.

Multiparameter multifractional brownian motion : local nondeterminism and joint continuity of the local times

Antoine Ayache;  Narn-Rueih Shieh;  Yimin Xiao — 2011

Annales de l'I.H.P. Probabilités et statistiques

By using a wavelet method we prove that the harmonisable-type -parameter multifractional brownian motion (mfBm) is a locally nondeterministic gaussian random field. This nice property then allows us to establish joint continuity of the local times of an (, )-mfBm and to obtain some new results concerning its sample path behavior.

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