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

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

2007-05-29

arxiv.org/abs/0705.4245v3

Mathematics

Probability

revised version

Scientific paper

Self-interacting diffusions are solutions to SDEs with a drift term depending on the process and its normalized occupation measure $\mu_t$ (via an interaction potential and a confinement potential). We establish a relation between the asymptotic behavior of $\mu_t$ and the asymptotic behavior of a deterministic dynamical flow (defined on the space of the Borel probability measures). We extend previous results on $\mathbb{R}^d$ or more generally a smooth complete connected Riemannian manifold without boundary. We will also give some sufficient conditions for the convergence of $\mu_t$. Finally, we will illustrate our study with an example on $\mathbb{R}^2$.

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