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2016 Self attracting diffusions on a sphere and application to a periodic case
Carl-Erik Gauthier
Electron. Commun. Probab. 21: 1-12 (2016). DOI: 10.1214/16-ECP4547

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 $c>0$.

Citation

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Carl-Erik Gauthier. "Self attracting diffusions on a sphere and application to a periodic case." Electron. Commun. Probab. 21 1 - 12, 2016. https://doi.org/10.1214/16-ECP4547

Information

Received: 10 September 2015; Accepted: 27 June 2016; Published: 2016
First available in Project Euclid: 1 August 2016

zbMATH: 1345.60087
MathSciNet: MR3533285
Digital Object Identifier: 10.1214/16-ECP4547

Subjects:
Primary: 60G17, 60J60, 60K35

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