Hausdorff–Besicovitch measure of fractal functional limit laws induced by Wiener process in Hölder norms

Abstract

Let { W ( t ) : t ⩾ 0 } denote a standard Wiener process. In this paper, we first establish a de Acosta [A. de Acosta, On the functional form of Lévy's modulus of continuity for Brownian motion, Z. Wahr. Verw. Gebiete 69 (1985) 567–579] type strong law for a family of Hölder norms. More precisely, we obtain, for α ∈ ( 0 , 1 / 2 ) , the exact rate of convergence, as h ↓ 0 , of T α , f ( h ) : = inf 0 ⩽ t ⩽ 1 − h ‖ ( 2 h log ( 1 / h ) ) − 1 / 2 ( W ( t + h ⋅ ) − W ( t ) ) − f ‖ α when f ∈ S satisfies ∫ 0 1 { d d u f ( u ) } 2 d u < 1 , where S denotes the Strassen [V. Strassen, An invariance principle for the law of the iterated logarithm, Z. Wahr. Verw. Gebiete 3 (1964) 211–226] set. In a second part we give some general technical tools for evaluating the upper and the lower critical functions of the Hausdorff–Besicovitch measures respectively for limsup random sets and for random Cantor type sets. As an application we deduce the Hausdorff dimension of the random fractal constituted of exceptional points in [ 0 , 1 ] where the previous rate is reached.