Nonlinear self-stabilizing processes – I Existence, invariant probability, propagation of chaos

Taking an odd, non-decreasing function β, we consider the (nonlinear) stochastic differential equation (Etilde)Xt=X0+Bt−12∫0tβ ∗ u(s,Xs)ds, t⩾0,P(Xt∈dx)=u(t,dx), t>0,and we prove the existence and uniqueness of solution of Eq. (E~), where β ∗ u(s,x)=∫Rβ(x−y)u(s, dy) and (Bt; t⩾0) is a one-dimensional Brownian motion, B0=0. We show that Eq. (E~)admits a stationary probability measure and investigate the link between Eq. (E~)and the associated system of particles.