  Via constructing an asymptotic coupling by reflection, in this paper we
establish uniform-in-time estimates on probability distances for mean-field
type SDEs, where the drift terms under consideration are dissipative merely in
the long distance. As applications, we (i) explore the long time probability
distance estimate between an SDE and its delay version; (ii) investigate the
issue on uniform-in-time propagation of chaos for McKean-Vlasov SDEs, where the
drifts might be singular with respect to the spatial variables and need not to
be of convolution type; (iii) tackle the discretization error bounds in an
infinite-time horizon for stochastic algorithms (e.g. backward/tamed/adaptive
Euler-Maruyama schemes as three typical candidates) associated with
McKean-Vlasov SDEs.
