McKean–Vlasov process

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 $\sigma :\mathbb {R} ^{d}\times {\mathcal {P}}(\mathbb {R} ^{d})\to {\mathcal {M}}_{d}(\mathbb {R} )$ where ${\mathcal {P}}(\mathbb {R} ^{d})$ is the space of probability distributions on $\mathbb {R} ^{d}$ equipped with the Wasserstein metric $W_{2}$ and ${\mathcal {M}}_{d}(\mathbb {R} )$ is the space of square matrices of dimension $d$. Consider a measurable function $b:\mathbb {R} ^{d}\times {\mathcal {P}}(\mathbb {R} ^{d})\to \mathbb {R} ^{d}$. Define $a(x,\mu ):=\sigma (x,\mu )\sigma (x,\mu )^{T}$.

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

• $X_{0}$ has law $f_{0}$
• $dX_{t}=\sigma (X_{t},\mu _{t})dB_{t}+b(X_{t},\mu _{t})dt$

where $\mu _{t}={\mathcal {L}}(X_{t})$ 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.

Existence of a solution — Suppose $b$ and $\sigma$ are globally Lipschitz, that is, there exists a constant $C>0$ such that:

$|b(x,\mu )-b(y,\nu )|+|\sigma (x,\mu )-\sigma (y,\nu )|\leq C(|x-y|+W_{2}(\mu ,\nu ))$

where $W_{2}$ is the Wasserstein metric.

Suppose $f_{0}$ has finite variance.

Then for any $T>0$ there is a unique strong solution to the McKean-Vlasov system of equations on $[0,T]$. Furthermore, its law is the unique solution to the non-linear Fokker–Planck equation:

$\partial _{t}\mu _{t}(x)=-\nabla \cdot \{b(x,\mu _{t})\mu _{t}\}+{\frac {1}{2}}\sum \limits _{i,j=1}^{d}\partial _{x_{i}}\partial _{x_{j}}\{a_{ij}(x,\mu _{t})\mu _{t}\}$

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 $(X_{t}^{i})_{1\leq i\leq N}$.

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

• $(X_{0}^{i})_{1\leq i\leq N}$ are i.i.d with law $f_{0}$
• $dX_{t}^{i}=\sigma (X_{t}^{i},\mu _{X_{t}})dB_{t}^{i}+b(X_{t}^{i},\mu _{X_{t}})dt$

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

Propagation of chaos is the property that, as the number of particles $N\to +\infty$, 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