Nonlinear self-stabilizing processes – II: Convergence to invariant probability

We now analyze the asymptotic behaviour of Xt, as t approaches infinity, X being solution of (1)Xt=X0+Bt−12∫0tb(s,Xs) ds,b(s,x)=E[β(x−Xs)],where β is a given odd and increasing Lipschitz-continuous function with polynomial growth. We prove with additional assumptions on β that Xt converges in distribution to the invariant probability measure associated with Eq. (1).