  We develop a new approach to study the long time behaviour of solutions to
nonlinear stochastic differential equations in the sense of McKean, as well as
propagation of chaos for the corresponding mean-field particle system
approximations. Our approach is based on a sticky coupling between two
solutions to the equation. We show that the distance process between the two
copies is dominated by a solution to a one-dimensional nonlinear stochastic
differential equation with a sticky boundary at zero. This new class of
equations is then analyzed carefully. In particular, we show that the
dominating equation has a phase transition. In the regime where the Dirac
measure at zero is the only invariant probability measure, we prove exponential
convergence to equilibrium both for the one-dimensional equation, and for the
original nonlinear SDE. Similarly, propagation of chaos is shown by a
componentwise sticky coupling and comparison with a system of one dimensional
nonlinear SDEs with sticky boundaries at zero. The approach applies to
equations without confinement potential and to interaction terms that are not
of gradient type.
