Système de processus auto-stabilisants
Volume 414 / 2003
Abstract
\def\BBr{{\sym R}}% \def\BBp{{\sym P}}Taking an odd increasing Lipschitz-continuous function with polynomial growth , an odd Lipschitz-continuous and bounded function \phi satisfying \sgn (x)\phi(x)\ge0 and a parameter a\in[1/2,1], we consider the (nonlinear) stochastic differential system \displaylines{ {\rm(E)}\indent \cases{ \displaystyle X_{t}=X_{0}+B_{t}+a\int_{0}^{t}\phi*v_{s}(X_{s})\,ds - (1-a)\int_{0}^{t}\beta*u_{s}(X_{s})\,ds, \cr \displaystyle Y_{t}=Y_{0}+ \widetilde{B}_{t}+(1-a)\int_{0}^{t}\phi*u_{s}(Y_{s})\,ds - a\int_{0}^{t}\beta*v_{s}(Y_{s})\,ds ,\cr}\hfill\cr \hfill \hfill\BBp(X_{t}\in dx )=u_{t}( dx )\quad \hbox{and}\quad \BBp(Y_{t}\in dx )=v_{t}(dx ),\indent\cr} where \beta*u_{t}(x)=\int_{\BBr}\beta(x-y)\,u_{t}(dy ), (B_{t})_{t\ge0} and (\widetilde{B}_{t})_{ t\ge0} are independent Brownian motions. We show that (E) admits a stationary probability measure, and, under some additional conditions, that (X_{t},Y_{t}) converges in distribution to this invariant measure. Moreover we investigate the link between (E) and the associated system of particles.