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Self-interacting diffusions: a simulated annealing version
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  • Published: 26 March 2008

Self-interacting diffusions: a simulated annealing version

  • Olivier Raimond1 

Probability Theory and Related Fields volume 144, pages 247–279 (2009)Cite this article

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  • 5 Citations

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Abstract

We study asymptotic properties of processes X, living in a Riemannian compact manifold M, solution of the stochastic differential equation (SDE)

$$dX_t = dW_t(X_t) - \beta(t)\nabla V\mu_t(X_t)dt$$

with W a Brownian vector field, β(t) = alog(t + 1), \(\mu_t = \frac{1}{t} \int_0^t \delta_{X_s}ds\) and \(V\mu_t(x) = \frac{1}{t}\int_0^t V(x, X_s)ds\), V being a smooth function. We show that the asymptotic behavior of μ t can be described by a non-autonomous differential equation. This class of processes extends simulated annealing processes for which V(x, y) is only a function of x. In particular we study the case \(M = {\mathbb{S}}^n\) , the n-dimensional sphere, and V(x, y) = −cos(d(x, y)), where d(x, y) is the distance on \({\mathbb{S}}^n\) , which corresponds to a process attracted by its past trajectory. In this case, it is proved that μ t converges almost surely towards a Dirac measure.

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Authors and Affiliations

  1. Département de Mathématiques, Université Paris-Sud, Bâtiment 425, 91405, Orsay cedex, France

    Olivier Raimond

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  1. Olivier Raimond
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Correspondence to Olivier Raimond.

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Raimond, O. Self-interacting diffusions: a simulated annealing version. Probab. Theory Relat. Fields 144, 247–279 (2009). https://doi.org/10.1007/s00440-008-0147-9

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  • Received: 30 October 2006

  • Revised: 06 February 2008

  • Published: 26 March 2008

  • Issue Date: May 2009

  • DOI: https://doi.org/10.1007/s00440-008-0147-9

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Mathematics Subject Classification (2000)

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