On the Besov regularity of the bifractional Brownian motion

Abstract

Our aim is to improve Hölder continuity results for the bifractional Brownian motion (bBm) (B α,β (t)) t∈[0,1] with 0 < α < 1 and 0 < β 1. We prove that almost all paths of the bBm belong to (resp. do not belong to) the Besov spaces Bes(αβ, p) (resp. bes(αβ, p)) for any 1 αβ < p < ∞, where bes(αβ, p) is a separable subspace of Bes(αβ, p). We also show similar regularity results in the Besov-Orlicz space Bes(αβ, M2) with M2(x) = e x 2 − 1. We conclude by proving the Itô-Nisio theorem for the bBm with αβ > 1/2 in the Hölder spaces C γ with γ < αβ.

On Besov regularity and local time of the solution to the stochastic heat equation

“…where C 1 , C 2 are two constants and (B HK (t)) t≥0 is a fBm with parameter HK and (X H,K (t)) t≥0 is a Gaussian process with infinitely differentiable trajectories on (0, +∞) and absolutely continuous on [0, +∞). On the other hand, we know from Ciesielski et al [15] (see also [10]) that almost all paths of the fBm (B HK (t)) t≥0 belong (resp. do not belong) to the Besov spaces B HK p (resp.…”

“…By using Theorem 1.4 and 1.5, the authors have widely investigated in [10] the Besov regularity of bBm, we mention the following theorem. Theorem 2.2 (Theorem 3.1 and 3.7 in [10]). Let (B H,K (t)) t∈[0,1] be a bBm with parameters H ∈ (0, 1) and K ∈ (0, 1].…”

“…Hence, we can not derive directly from Lei and Nualart decomposition the Besov regularity, when x ∈ R is fixed, of (u(t, x) , t ∈ [0, ε]) or more generally the Besov regularity of the bBm on the interval [0, ε]. Therefore, To fill this gap, the authors have investigated in [10] the Besov regularity for sample paths of the bBm (B H,K (t)) for t ∈ [0, 1].…”

“…See section 1.3 for more details and [48] for an introduction to Besov spaces. In [10], the authors have investigated the Besov regularity of the bifractional Brownian motion. However, we know that, for each fixed x ∈ R, the Gaussian process (u(t, x) ; t ≥ 0) is a bifractional Brownian motion (bBm for short) with parameters H = K = 1 2 , multiplied by a constant (see Remark 6).…”