Self attracting diffusions on a sphere and application to a periodic case

Abstract

This paper proves almost-sure convergence for the self-attracting diffusion on the unit spherey is the usual scalar product in R n , and (Wt(.)) t 0 is a Brownian motion on S n . From this follows the almost-sure convergence of the real-valued self-attracting diffusion dϑt = σdWt + a t 0 sin(ϑt − ϑs)dsdt, where (Wt) t 0 is a real Brownian motion.keywords: reinforced process, self-interacting diffusions, asymptotic pseudotrajectories, rate of convergence.MSC 60K35, 60G17, 60J60

Strongly self-interacting processes on the circle

“…Later, S.Herrmann and B.Roynette extended this result to a broader class of potentials of the form W (x, y) = V (x − y) with V convex (see [12]). In the case of the circle, the first author obtained the same result (almost sure convergene toward a random variable) for the interaction potential W (x, y) = − cos(x − y) (see [9]). In all these cases the particle is attracted by its past.…”

Least squares estimation for a linear self-repelling diffusion driven by fractional Brownian motion

“…(ii) 常自交互情形 [4][5][6][7], 关于一般自交互扩散的研究可参见文献 [8][9][10][11]. 最近, 受文献 [12,13] 的启发, Yan 等 [14] 考虑了随机微分方程…”

Large Time Behavior on the Linear Self-Interacting Diffusion Driven by Sub-Fractional Brownian Motion With Hurst Index Large Than 0.5 I: Self-Repelling Case

“…A great difference between these diffusions and Brownian polymers is that the drift term is divided by t. It is noteworthy that the interaction potential is attractive enough to compare the diffusion (a bit modified) to an Ornstein-Uhlenbeck process, in many case of f, which points out an access to its asymptotic behavior. More works can be found in Benaïm et al [5], Cranston and Mountford [6], Gauthier [7], Herrmann and Roynette [8], Herrmann and Scheutzow [9], Mountford and Tarr [10], Shen et al [11], Sun and Yan [12] and the references therein.…”