  In this paper, by introducing a new type asymptotic coupling by reflection,
we explore the long time behavior of random probability measure flows
associated with a large class of one-dimensional McKean-Vlasov SDEs with common
noise. Concerning the McKean-Vlasov SDEs with common noise under consideration
in the present work, in contrast to the existing literature, the drift terms
are much more general rather than of the convolution form, and, in particular,
can be of polynomial growth with respect to the spatial variables, and moreover
idiosyncratic noises are allowed to be of multiplicative type. Most
importantly, our main result indicates that both the common noise and the
idiosyncratic noise facilitate the exponential contractivity of the associated
measure-valued processes.
