On the Besov regularity of the bifractional Brownian motion

Abstract

Our aim in this paper is to improve H\"{o}lder continuity results for the bifractional Brownian motion (bBm) $(B^{\alpha,\beta}(t))_{t\in[0,1] }$ with $0<\alpha<1$ and $0<\beta\leq 1$. 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 It\^{o}-Nisio theorem for the bBm with $\alpha \beta>\frac{1}{2}$ in the H\"{o}lder spaces $\mathcal{C}^{\gamma}$, with $\gamma<\alpha \beta$.