Diffusion process

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A continuous Markov process with transition density which satisfies the following condition: There exist functions and , known as the drift coefficient and the diffusion coefficient respectively, such that for any ,


it being usually assumed that these limit relations are uniform with respect to in each finite interval and with respect to , . An important representative of this class of processes is the process of Brownian motion, which was originally considered as a mathematical model of diffusion processes (hence the name "diffusion process" ).

If the transition density is continuous in and together with its derivatives and , it is the fundamental solution of the differential equation


which is known as the backward Kolmogorov equation (cf. also Kolmogorov equation).

In the homogeneous case, when the drift coefficient and the diffusion coefficient are independent of the time , the backward Kolmogorov equation for the respective transition density has the form

If the transition density has a continuous derivative in and such that the functions and are continuous in , it is the fundamental solution of the differential equation


known as the Fokker–Planck equation, or the forward Kolmogorov equation. The differential equations (2) and (3) for the probability density are the fundamental analytic objects of study of diffusion processes. There is also another, purely "probabilistic" , approach to diffusion processes, based on the representation of the process as the solution of the Itô stochastic differential equation

where is the standard process of Brownian motion. Roughly speaking, is considered to be connected with some Brownian motion process in such a way that if , then the increment during the next period of time is

If this asymptotic relation is understood in the sense that

where are magnitudes of the same type as in equations (1), the under consideration will constitute a diffusion process in the sense of this definition as well.

Multi-dimensional diffusion process is the name usually given to a continuous Markov process in an -dimensional vector space whose transition density satisfies the following conditions: For any ,

The vector characterizes the local drift of the process , and the matrix , , characterizes the mean square deviation of the random process from the initial position in a small period of time between and .

Subject to certain additional restrictions, the transition density of a multi-dimensional diffusion process satisfies the forward and backward Kolmogorov differential equations:

A multi-dimensional diffusion process may also be described with the aid of Itô's stochastic differential equations:

where are mutually-independent Brownian motion processes, while

are the eigen vectors of the matrix .


[1] I.I. Gikhman, A.V. Skorokhod, "Introduction to the theory of random processes" , Saunders (1969) (Translated from Russian)
[2] I.I. Gikhman, A.V. Skorokhod, "Stochastic differential equations and their applications" , Springer (1972) (Translated from Russian)


Instead of backward Kolmogorov equation and forward Kolmogorov equation are also finds simply backward equation and forward equation.


[a1] N. Ikeda, S. Watanabe, "Stochastic differential equations and diffusion processes" , North-Holland & Kodansha (1981)
[a2] D.W. Stroock, S.R.S. Varadhan, "Multidimensional diffusion processes" , Springer (1979)
[a3] L. Arnold, "Stochastische Differentialgleichungen" , R. Oldenbourg (1973) (Translated from Russian)
How to Cite This Entry:
Diffusion process. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by Yu.A. Rozanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article