Abstract
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.
Highlights
Reviewed by: Yu Sun, Our Lady of the Lake University, United States Zhenxia Liu, Linköping University, Sweden
Specialty section: This article was submitted to Interdisciplinary Physics, a section of the journal Frontiers in Physics
A great difference between these diffusions and Brownian polymers is that the drift term is divided by t
Summary
We briefly recall the definition and properties of stochastic integral with respect to sub-fBm. Throughout this paper we assume that SH SHt , t ≥ 0 denotes a sub-fBm defined on the probability space (Ω, F , P) with index H. The estimate (1.6) and normality imply that the sub-fBm t1SHt admits almost surely a bounded H1−θ-variation on any finite interval for any sufficiently small θ ∈ (0, H). One can define the Young integral of a process u {ut, t ≥ 0} with respect to subfBm Ba,b t usdSHs as the limit in probability of a Riemann sum. The integral is well-defined and utSHt t usdSHs + SHs dus for all t ≥ 0, provided u is of bounded qH-variation on any finite interval qH. We will denote δ(u) usδSHs for an adapted process u, and it is called Skorohod integral.
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