probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

, a McKean–Vlasov process is a

stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...

described by a

stochastic differential equation A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock pr ...

where the coefficients of the

diffusion Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...

depend on the distribution of the solution itself. The equations are a model for

Vlasov equation The Vlasov equation is a differential equation describing time evolution of the Distribution function (physics), distribution function of plasma (physics), plasma consisting of charged particles with long-range interaction, e.g. Coulomb's law, Coulo ...

and were first studied by

Henry McKean Henry P. McKean, Jr. (born 1930 in Wenham, Massachusetts) is an American mathematician at the Courant Institute in New York University. He works in various areas of mathematical analysis, analysis. He obtained his Doctor of Philosophy, PhD in 1955 ...

in 1966.

Definition

McKean–Vlasov processes take the form

$dX_t = a(X_t, \mu_t) dB_t + b(X_t, \mu_t) dt$

where

\mu_t = \mathcal(X_t)

describes the

law Law is a set of rules that are created and are enforceable by social or governmental institutions to regulate behavior,Robertson, ''Crimes against humanity'', 90. with its precise definition a matter of longstanding debate. It has been vario ...

of X and dB denotes the

Wiener process In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is o ...

. That is the coefficients of the SDE depend on the marginal distribution of the process X. In general, the process can describe non-linear diffusion.

Applications

Mean-field theory In physics and probability theory, Mean-field theory (MFT) or Self-consistent field theory studies the behavior of high-dimensional random (stochastic) models by studying a simpler model that approximates the original by averaging over degrees of ...

Mean-field game theory Mean-field game theory is the study of strategic decision making by small interacting agents in very large populations. It lies at the intersection of game theory with stochastic analysis and control theory. The use of the term "mean field" is insp ...

Random matrices In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all elements are random variables. Many important properties of physical systems can be represented mathemat ...

: including Dyson's model on eigenvalue dynamics for random symmetric matrices and the

Wigner semicircle distribution The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution on minus;''R'', ''R''whose probability density function ''f'' is a scaled semicircle (i.e., a semi-ellipse) centered at (0, 0): :f(x)=\sq ...

References

Stochastic differential equations {{probability-stub