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Abstract

\def\BBr{{\sym R}}% \def\BBp{{\sym P}}Taking an odd increasing Lipschitz-continuous function with polynomial growth $\beta$, 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.