Self-interacting diffusions.

*(English)*Zbl 1042.60060The 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.

Reviewer: Martin Janžura (Praha)