Self-interacting diffusions

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• 作者：Michel Bena?m ; Michel Ledoux and Olivier Raimond
• 刊名：Probability Theory and Related Fields
• 出版年：2002
• 出版时间：January 2002
• 年：2002
• 卷：122
• 期：1
• 页码：1-41
• 全文大小：338 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Probability Theory and Stochastic Processes
  Mathematical and Computational Physics
  Quantitative Finance
  Mathematical Biology
  Statistics for Business, Economics, Mathematical Finance and Insurance
  Operation Research and Decision Theory
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-2064

文摘

?This paper is concerned with a general class of self-interacting diffusions {X _t } _{t ≥0} living on a compact Riemannian manifold M. These are solutions to stochastic differential equations of the form : dX _t = Brownian increments + drift term depending on X _t and μ _t , the normalized occupation measure of the process. It is proved that the asymptotic behavior of {μ _t } can be precisely related to the asymptotic behavior of a deterministic dynamical semi-flow Φ = {Φ _t } _{t ≥0} defined on the space of the Borel probability measures on M. In particular, the limit sets of {μ _t } are proved to be almost surely attractor free sets for Φ. These results are applied to several examples of self-attracting/repelling diffusions on the n-sphere. For instance, in the case of self-attracting diffusions, our results apply to prove that {μ _t } can either converge toward the normalized Riemannian measure, or to a gaussian measure, depending on the value of a parameter measuring the strength of the attraction.