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
