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Publication# Propriétés fines des trajectoires du mouvement brownien fractionnaire

Abstract

Let X = {X(t); t ∈ RN} be a (N,d) fractional Brownian motion in Rd of index H ∈ (0,1). We study the local time of X for all temporal dimensions N and spatial dimensions d for which local time exist. We obtain two main results : R1. If we denote by Lx(I) the local time of X at x on I ⊂ RN, then there exists a positive finite constant c such that mφ(X-1(0) ∩ [0,1]N) = c L0([0,1]N), where φ(r) = rN-dH (log log 1/r)dH/N and mφ(E) is the Hausdorff φ-measure of E. This solves the problem of the relationship between the local time and the exact Hausdorff measure of zero set for X. R2. We refine results of Xiao (1997) for the local times of (N,d) fractional Brownian motion. We prove the law of iterated logarithm and global Hölder condition for the local time for our process. These results establish interesting properties which were only partially proved in the literature. This literature began with the work of Taylor and Wendel (1966) and Perkins (1981) for the first result; Kesten (1965) and Perkins (1981) for the second on the Brownian motion. It continued with several works of which that of Xiao (1997) on the locally nondeterministic processes with stationnary increments including the (N,d) fractional Brownian motion. An intermediate result is found to solve the case N > 1. We generalize the result of Kasahara et al. (1999) on the tail probability of local time.

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