Let SH be a sub-fractional Brownian motion with index 12<H<1. In this paper, we consider the linear self-interacting diffusion driven by SH, which is the solution to the equationdXtH=dStH−θ(∫0tXtH−XsHds)dt+νdt,X0H=0,where θ &lt; 0 and ν∈R are two parameters. Such process XH is called self-repelling and it is an analogue of the linear self-attracting diffusion [Cranston and Le Jan, Math. Ann. 303 (1995), 87–93]. Our main aim is to study the large time behaviors. We show the solution XH diverges to infinity, as t tends to infinity, and obtain the speed at which the process XH diverges to infinity as t tends to infinity.

In this study, as a continuation to the studies of the self-interaction diffusion driven by subfractional Brownian motion SH, we analyze the asymptotic behavior of the linear self-attracting diffusion:dXtH=dStH−θ∫0t(XtH−XsH)dsdt+νdt,X0H=0,where θ &gt; 0 and ν∈R are two parameters. When θ &lt; 0, the solution of this equation is called self-repelling. Our main aim is to show the solution XH converges to a normal random variable X∞H with mean zero as t tends to infinity and obtain the speed at which the process XH converges to X∞H as t tends to infinity.

Correction for ‘LAPONITE® nanodisk-based platforms for cancer diagnosis and therapy’ by Gaoming Li et al., Mater. Adv., 2022, https://doi.org/10.1039/d2ma00637e.

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Let B = B t 1 , … , B t d t ≥ 0 be a d -dimensional bifractional Brownian motion and R t = B t 1 2 + ⋯ + B t d 2 be the bifractional Bessel process with the index 2 HK ≥ 1 . The Itô formula for the bifractional Brownian motion leads to the equation R t = ∑ i = 1 d ∫ 0 t B s i / R s d B s i + HK d − 1 ∫ 0 t s 2 HK − 1 / R s d s . In the Brownian motion case K = 1 and H = 1 / 2 , X t ≔ ∑ i = 1 d ∫ 0 t B s i / R s d B s i ,   d ≥ 1 is a Brownian motion by Lévy’s characterization theorem. In this paper, we prove that process X t is not a bifractional Brownian motion unless K = 1 and H = 1 / 2 . We also study some other properties and their application of this stochastic process.