On the Besov regularity of the bifractional Brownian motion

Abstract

Our aim in this paper is to improve Hölder continuity results for the bifractional Brownian motion (bBm) $(B^{\alpha ,\beta }(t){)}_{t\in [0,1]}$ with $0<\alpha <1$ and $0<\beta \le 1$. We prove that almost all paths of the bBm belong (resp. do not belong) to the Besov spaces $\mathbf{Bes}(\alpha \beta ,p)$ (resp. $\mathbf{bes}(\alpha \beta ,p)$) for any $\frac{1}{\alpha \beta }<p<\infty$, where $\mathbf{bes}(\alpha \beta ,p)$ is a separable subspace of $\mathbf{Bes}(\alpha \beta ,p)$. We also show the Itô-Nisio theorem for the bBm with $\alpha \beta >\frac{1}{2}$ in the Hölder spaces ${\mathcal{C}}^{\gamma }$, with $\gamma <\alpha \beta$.

20 pages