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 sphere $$dX(t)=\sigma d{W}_t(X(t))-a{\int }_0^t{\nabla }_{{\mathbb{S}}^n}{V}_{X_s}(X_t)dsdt,X(0)=x\in {\mathbb{S}}^n$$ %given by the stochastic differential equation: $$d{X}_t=\sigma d{W}_t+a{\int }_0^t\sin (X_t-X_s)dsdt,$$ where $\sigma >0$, $a<0$, $V_y(x)=⟨x,y⟩$ is the usual scalar product in ${\mathbb{R}}^n$, and $(W_t(.){)}_{t⩾0}$ is a Brownian motion on ${\mathbb{S}}^n$. From this follows the almost-sure convergence of the real-valued self-attracting diffusion $$d{\vartheta }_t=\sigma d{W}_t+a{\int }_0^t\sin ({\vartheta }_t-{\vartheta }_s)dsdt,$$ where $(W_t{)}_{t⩾0}$ is a real Brownian motion.

arXiv commentVersion 1: 15 pages. Version 2: The result is extended to the case of the n-dimensional unit sphere. The proofs were adapted and improved, the presentation is made more transparent, but the guideline remains identical. Therefore the title was changed