2015/01/20 by Carl-Erik Gauthier, Gauthier, Carl-Erik
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR
paper · pdf · doi:10.48550/arxiv.1501.04827
Version 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
arxiv created 2015/09/04 · arxiv updated 2015/09/07
This paper proves almost-sure convergence for the self-attracting diffusion on the unit sphere dX(t)=σdWt(X(t))-a∫0t∇_\mathbbSnVXs(Xt) dsdt, X(0)=x∈\mathbbSn %given by the stochastic differential equation: dXt=σdWt+a∫0tsin(Xt-Xs)dsdt, where σ>0, a < 0, Vy(x)=⟨ x,y⟩ is the usual scalar product in ℝn, and (Wt(.))t\geqslant 0 is a Brownian motion on \mathbbSn. From this follows the almost-sure convergence of the real-valued self-attracting diffusion dϑt=σdWt+a∫0tsin(ϑt-ϑs)dsdt, where (Wt)t\geqslant 0 is a real Brownian motion.