McKean–Vlasov process

Template:Short description 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. The equations are a model for Vlasov equation and were first studied by Henry McKean in 1966. 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.

Definition

Consider a measurable function $\sigma :{ℝ}^d\times 𝒫({ℝ}^d)\to {ℳ}_d(ℝ)$ where $𝒫({ℝ}^d)$ is the space of probability distributions on ${ℝ}^d$ equipped with the Wasserstein metric $W_2$ and ${ℳ}_d(ℝ)$ is the space of square matrices of dimension $d$. Consider a measurable function $b:{ℝ}^d\times 𝒫({ℝ}^d)\to {ℝ}^d$. Define $a(x,\mu ):=\sigma (x,\mu )\sigma (x,\mu {)}^T$.

A stochastic process $(X_t{)}_{t\ge 0}$ is a McKean–Vlasov process if it solves the following system:

• $X_0$ has law $f_0$
• $d{X}_t=\sigma (X_t,{\mu }_t)d{B}_t+b(X_t,{\mu }_t)dt$

where ${\mu }_t=ℒ(X_t)$ describes the law of $X$ and $B_t$ denotes a $d$-dimensional Wiener process. This process is non-linear, in the sense that the associated Fokker-Planck equation for ${\mu }_t$ is a non-linear partial differential equation.

Existence of a solution

The following Theorem can be found in.

Propagation of chaos

The McKean-Vlasov process is an example of propagation of chaos. What this means is that many McKean-Vlasov process can be obtained as the limit of discrete systems of stochastic differential equations $(X_t^i{)}_{1\le i\le N}$.

Formally, define $(X^i{)}_{1\le i\le N}$ to be the $d$-dimensional solutions to:

• $(X_0^i{)}_{1\le i\le N}$ are i.i.d with law $f_0$
• $d{X}_t^i=\sigma (X_t^i,{\mu }_{X_t})d{B}_t^i+b(X_t^i,{\mu }_{X_t})dt$

where the $(B^i{)}_{1\le i\le N}$ are i.i.d Brownian motion, and ${\mu }_{X_t}$ is the empirical measure associated with $X_t$ defined by ${\mu }_{X_t}:=\frac{1}{N}\sum _{1\le i\le N}{\delta }_{X_t^i}$ where $\delta$ is the Dirac measure.

Propagation of chaos is the property that, as the number of particles $N\to +\mathrm{\infty }$, the interaction between any two particles vanishes, and the random empirical measure ${\mu }_{X_t}$ is replaced by the deterministic distribution ${\mu }_t$.

Under some regularity conditions, the mean-field process just defined will converge to the corresponding McKean-Vlasov process.

Applications

• Mean-field theory
• Mean-field game theory
• Random matrices: including Dyson's model on eigenvalue dynamics for random symmetric matrices and the Wigner semicircle distribution