Inprobability theory, aMcKean–Vlasov process is astochastic process described by astochastic differential equation where the coefficients of thediffusion depend on the distribution of the solution itself.^[1][2] The equations are a model forVlasov equation and were first studied byHenry McKean in 1966.^[3] It is an example ofpropagation 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.^[4]

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

A stochastic process${\displaystyle (X_t{)}_{t\ge 0}}$ is a McKean–Vlasov process if it solves the following system:^[3][5]

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

where${\displaystyle {\mu }_t=\mathcal{L}(X_t)}$ describes thelaw of${\displaystyle X}$ and${\displaystyle B_t}$ denotes a${\displaystyle d}$-dimensionalWiener process. This process is non-linear, in the sense that the associatedFokker-Planck equation for${\displaystyle {\mu }_t}$ is a non-linearpartial differential equation.^[5][6]

The following Theorem can be found in.^[4]

The McKean-Vlasov process is an example ofpropagation 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${\displaystyle (X_t^i{)}_{1\le i\le N}}$.

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

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

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

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

Under some regularity conditions,^[4] the mean-field process just defined will converge to the corresponding McKean-Vlasov process.

• Mean-field theory
• Mean-field game theory^[5]
• Random matrices: including Dyson's model on eigenvalue dynamics for random symmetric matrices and theWigner semicircle distribution^[6]

• Stochastic differential equations

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