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Self-interacting diffusions. (English) Zbl 1042.60060
The authors are concerned with a general class of self-interacting diffusion processes. These are continuous time stochastic processes living on a compact Riemannian manifold $$M$$ which can be typically described as solutions to a stochastic differential equation of the form $dX_t=dW_t(X_t)-{1\over t} \left(\int_0^t \triangledown V_{X_s} (X_t)\,d s\right)d t$ where $${W_t}$$ is a Brownian vector field on $$M$$ and $$V_u (x)$$ a “potential” function. Here, the drift term depends on the normalized occupation measure $$\mu_t = {1\over t} \int_0^t \delta_{X_s} d s$$ of the process. The main goal of the paper is to give a systematic treatment of this class of processes and to describe with a great deal of generality the asymptotic behaviour of $$\mu_t$$ as $$t\rightarrow\infty$$. It is proved that the asymptotic behaviour of $$\mu_t$$ can be precisely related to the asymptotic behaviour of a deterministic dynamical semi-flow $$\Phi={\{\Phi_t\}}_{t\geq0}$$ defined on the space of the Borel probability measures on $$M$$. In particular, the limit sets of $$\mu_t$$ are proved to be almost surely attractor free sets for $$\Phi$$. These results are applied to several examples of self-attracting/repelling diffusions on the $$n$$-sphere. For instance, in case of self-attracting diffusions, $$\mu_t$$ can either converge towards the normalized Riemannian measure, or to a Gaussian measure, depending on the value of the parameter measuring the strength of the attraction. A numerical study is also included.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 58J65 Diffusion processes and stochastic analysis on manifolds 60J60 Diffusion processes
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