Packing measure of the sample paths of fractional Brownian motion

Abstract

Let X ( t ) ( t ∈ R ) X(t) (t \in \mathbf {R}) be a fractional Brownian motion of index α \alpha in R d . \mathbf {R}^d. If 1 > α d 1 > \alpha d , then there exists a positive finite constant K K such that with probability 1, \[ ϕ − p ( X ( [ 0 , t ] ) ) = K t for any t > 0 , \phi -p(X([0,t])) = Kt\quad \text {for any $t > 0$}, \] where ϕ ( s ) = s 1 α / ( log ⁡ log ⁡ 1 s ) 1 2 α \phi (s) = s^{\frac 1 { \alpha }}/ (\log \log \frac 1 s)^{\frac 1 {2 \alpha }} and ϕ \phi - p ( X ( [ 0 , t ] ) ) p (X([0,t])) is the ϕ \phi -packing measure of X ( [ 0 , t ] ) X([0,t]) .