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 \[ dX_t=\nu \circ dW_{t}(X_t)-a\int _{0}^{t}\nabla _{\mathbb{S} ^n}V_{X_s}(X_t) dsdt,\qquad X_0=x\in \mathbb{S} ^n, \] where $\nu >0$, $a < 0$, $V_y(x)=\langle x,y\rangle $ is the usual scalar product on $\mathbb{R} ^{n+1}$, $\circ $ stands for the Stratonovich differential and $(W_{t}(.))_{t\geqslant 0}$ is a Brownian vector field on $\mathbb{S} ^n$. From this we deduce the almost-sure convergence of the real-valued self attracting diffusion \[ d\vartheta _{t}=\nu dW_{t}+a\int _{0}^{t}\sin (c(\vartheta _{t}-\vartheta _{s}))dsdt, \] where $(W_t)_{t\geqslant 0}$ is a real Brownian motion and $c>0$.