The ODE method for some self-interacting diffusions on non-compact spaces

2007/05/29 by A. Kurtzmann, Kurtzmann, A.
Mathematics · #37C50 #60K35 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:37C50 #msc:60K35

paper · pdf · doi:10.48550/arxiv.0705.4245

revised version

arxiv created 2008/02/17 · arxiv updated 2009/12/01

Abstract

Self-interacting diffusions are solutions to SDEs with a drift term depending on the process and its normalized occupation measure μ_t (via an interaction potential and a confinement potential). We establish a relation between the asymptotic behavior of μ_t and the asymptotic behavior of a deterministic dynamical flow (defined on the space of the Borel probability measures). We extend previous results on ℝ^d or more generally a smooth complete connected Riemannian manifold without boundary. We will also give some sufficient conditions for the convergence of μ_t. Finally, we will illustrate our study with an example on ℝ².

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