In probability theory, a McKean–Vlasov process is a stochastic process described by a stochastic differential equation where the coefficients of the diffusion depend on the distribution of the solution itself.
[1][2] The equations are a model for Vlasov equation and were first studied by Henry McKean in 1966.
[3] It is an example of propagation of chaos, in that it can be obtained as a limit of a mean-field system of interacting particles: as the number of particles tends to infinity, the interactions between any single particle and the rest of the pool will only depend on the particle itself.
is the space of probability distributions on
equipped with the Wasserstein metric
is the space of square matrices of dimension
Consider a measurable function
A stochastic process
is a McKean–Vlasov process if it solves the following system:[3][5] where
describes the law of
-dimensional Wiener process.
This process is non-linear, in the sense that the associated Fokker-Planck equation for
is a non-linear partial differential equation.
[5][6] The following Theorem can be found in.
[4] Existence of a solution — Suppose
are globally Lipschitz, that is, there exists a constant
is the Wasserstein metric.
has finite variance.
there is a unique strong solution to the McKean-Vlasov system of equations on
Furthermore, its law is the unique solution to the non-linear Fokker–Planck equation:
The McKean-Vlasov process is an example of propagation of chaos.
[4] What this means is that many McKean-Vlasov process can be obtained as the limit of discrete systems of stochastic differential equations
Formally, define
are i.i.d Brownian motion, and
is the empirical measure associated with
is the Dirac measure.
Propagation of chaos is the property that, as the number of particles
, the interaction between any two particles vanishes, and the random empirical measure
is replaced by the deterministic distribution
Under some regularity conditions,[4] the mean-field process just defined will converge to the corresponding McKean-Vlasov process.