Global Dynamics of McKean-Vlasov SDEs via Stochastic Order

Abstract: This paper studies the rich dynamics of general one-dimensional McKean-Vlasov stochastic differential equations based on the preservation of stochastic order. The existing work is limited to the case of additive noises, while our framework allows multiplicative noises. When the equation has multiple invariant measures, we show basins of attraction of certain invariant measures contain unbounded open sets in the 2-Wasserstein space, which is vacant in previous research even for additive noises. Also, our main results target on total orderedness and finiteness of invariant measures, global convergence to order interval enclosed by two order-related invariant measures, alternating arrangement of invariant measures in terms of stability (locally attracting) and instability (connecting orbits). Our theorems cover a wide range of classical granular media equations, such as symmetric, flat-bottom and asymmetric double-well confinement potentials with quadratic interaction, multi-well landscapes, and double-well landscapes with perturbations. Specific values for the parameter ranges, explicit descriptions of attracting sets and phase diagrams are provided.