On the McKean-Vlasov dynamics with or without common noise
Hammersley, William Richard Philip
McKean-Vlasov stochastic differential equations may arise from a probabilistic interpretation of certain non-linear PDEs or as the limiting behaviour of mean field particle systems (those whose interactions are through the empirical measure) as the population size increases to infinity. Interest in this topic has grown enormously in recent times following the introduction of the related mean field games. These are models derived from the infinite population limit of games with finitely many players and mean field structure, i.e. the dynamics and rewards of one player depend on the other players through the empirical measure. Naturally, it is imperative that the dynamics of the models are well-posed. This question comprises the majority of this text in two stochastic contexts: with or without a common noise. In the more often studied case where the particles are driven by independent Brownian motions, results are provided that pertain to the weak-existence and pathwise continuous dependence on the initial condition. These results adapt a method of Gyöngy and Krylov for Itô's stochastic differential equations to the McKean-Vlasov setting. Should the coefficients and initial distribution satisfy a certain Lyapunov condition, well-posedness of the dynamics may be established along with the existence of an invariant measure for an associated semi-group. These conditions allow for potentially unbounded coefficients, with growth intrinsically linked to the Lyapunov condition. In the second context, particle systems driven by correlated noises are considered. In particular, the particles are each driven by two Brownian motions: one common to all particles and a private Brownian motion independent of all others. The connection between these particle systems and related McKean-Vlasov models through the conditional propagation of chaos is discussed. Existence and uniqueness of weak solutions to the corresponding McKean-Vlasov dynamics is proved in a particular framework that allows for a discontinuous drift coefficient at a price of non-degenerate noise.