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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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