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On self attracting/repelling diffusions. (English) Zbl 1030.60075
A self-interacting diffusion is a continuous stochastic process which resolves the SDE of the form $dX_t= \sum_i F_i(X_t)\circ dB^i_t- {\alpha\over 2t} \Biggl(\int^t_0\nabla V_{X_s}(X_t) ds\Biggr) dt$ on a compact Riemannian manifold $$M$$, where $$(u,x)\to V_u(x)$$ is a smooth function on $$M\times M$$. The main purpose of this work is to study the empirical occupation measure $$\mu_t={1\over t} \int^t_0 \delta_{X_s}ds$$. In the case where $$V$$ has the form $V(x,y)= \int_C G(u,x) G(u,y)\nu(du)+ \beta,$ it is shown that the limit set $$L\{\mu_t\}$$ of $$\{\mu_t; t> 0\}$$ is a connected subset in the space of Borel probability measures $${\mathcal P}(M)$$ over $$M$$. Moreover any $$\mu\in L\{\mu_t\}$$ admits a density which was characterized by an “energy function”. A sufficient condition on $$\alpha$$ is given to insure that $$\mu^*= \lim_{t\to+\infty} \mu_t$$ exists almost surely.

##### MSC:
 60J60 Diffusion processes 60F15 Strong limit theorems
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##### References:
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