McKean–Vlasov SDEs under measure dependent Lyapunov conditions

Abstract

We prove the existence of weak solutions to McKean–Vlasov SDEs defined on a domain D⊆Rd with continuous and unbounded coefficients and degenerate diffusion coefficient. Using differential calculus for the flow of probability measures due to Lions, we introduce a novel integrated condition for Lyapunov functions in an infinite dimensional space D×P(D), where P(D) is a space of probability measures on D. Consequently we show existence of solutions to the McKean–Vlasov SDEs on [0,∞). This leads to a probabilistic proof of the existence of a stationary solution to the nonlinear Fokker–Planck–Kolmogorov equation under very general conditions. Finally, we prove uniqueness under an integrated condition based on a Lyapunov function. This extends the standard monotone-type condition for uniqueness.