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

Taking an odd, non-decreasing function beta, we consider the (nonlinear) stochastic differential equation X-t = X-0 + B-t - 1/2 integral(0)(t) beta * u(s, X-s)ds, t greater than or equal to 0, P(X-t is an element of dx) = u(t, dx), t>0 ((E) over tilde) and we prove the existence and uniqueness of solution of Eq. ((E) over tilde), where beta * u(s,x) = integral(R) beta(x-y)u(s, dy) and (B-t; t greater than or equal to 0) is a one-dimensional Brownian motion, B-0 = 0. We show that Eq. ((E) over tilde) admits a stationary probability measure and investigate the link between Eq, ((E) over tilde) and the associated system of particles. (C) 1998 Published by Elsevier Science B.V. All rights reserved.