Self-attracting diffusions: Case of the constant interaction
The paper deals with a stochastic equation of the form \[ dX_t= dB_t-\sigma \int^t_0 {X_t-X_s \over |X_t- X_s|} ds dt \] with \(X_0=0\), where \(\sigma\) is a positive constant and \(B_t\) is a \(d\)-dimensional Brownian motion. This equation describes a diffusion process attracted by its own trajectories. It has been studied before by M. Cranston and Y. Le Jan in the one-dimensional case. In the present paper the general \(d\)-dimensional case is considered. The main theorem states that the solution \(X_t\) converges almost surely. The proof requires new ideas with respect to the one-dimensional case, and in particular a good understanding of the action of the restoring drift.
scientific article; zbMATH DE number 1002143