# Ornstein-Uhlenbeck process

A Gaussian stationary random process $V( t)$ with zero expectation and an exponentially damped correlation function of the form

$${\mathsf E} V( t) V( t + \tau ) = \ B( \tau ) = \sigma ^ {2} \mathop{\rm exp} (- \alpha | \tau | ),\ \alpha > 0.$$

An Ornstein–Uhlenbeck process can also be defined as a stationary solution of the stochastic equation (Langevin equation):

$$\tag{* } m dV( t) + \beta V( t) dt = dW( t),$$

where $W( t)$ is a Wiener process (i.e. a process for which $dW( t)/dt = W ^ \prime ( t)$ is a white noise process), while $m$ and $\beta$ are positive constants with $\beta /m = \alpha$.

Equation (*) approximately describes a one-dimensional Brownian motion of a free particle in a fluid; $V( t)$ is here interpreted as the velocity of the particle, $m$ is its mass, $- \beta V( t)$ is the force of "viscous friction" proportional to the velocity (for a spherical particle of radius $a$, the coefficient $\beta$ is equal to $6 \pi \eta a$, where $\eta$ is the fluid's viscosity, by virtue of Stokes' fluid hydrodynamic law), while the white noise $W ^ \prime ( t)$ is a "random force" , which is generated by chaotic shocks from the fluid molecules in thermal motion, and is the basic cause of the Brownian motion. In the original theory of Brownian motion, developed by A. Einstein and M.V. Smoluchowski in 1905–1906, the inertia of the particle was disregarded, i.e. $m$ was taken to be equal to 0; equation (*) then led to the conclusion that the coordinate of a Brownian particle

$$X( t) = \int\limits _ { 0 } ^ { t } V( t ^ \prime ) dt ^ \prime$$

is equal to $\beta ^ {-1} W( t)$, i.e. is a Wiener process. The Wiener process thus describes the Einstein–Smoluchowski model of Brownian motion (hence its other name — Brownian motion process); since this process is non-differentiable, a Brownian particle in the Einstein–Smoluchowski theory does not have a finite velocity. The refined Brownian motion theory, which relies on equation (*) where $m \neq 0$, was proposed by L.S. Ornstein and G.E. Uhlenbeck (; see also ); the same theory was subsequently put forward also by S.N. Bernshtein  and A.N. Kolmogorov . In the Ornstein–Uhlenbeck theory, the velocity $V( t)$ of the Brownian particle is finite, but its acceleration is infinite (since the Ornstein–Uhlenbeck process is non-differentiable); for the acceleration to be finite, the theory must be further refined by taking into account the fact that a random force differs from an idealized white noise process $W ^ \prime ( t)$.

Equation (*) can also be used to describe the one-dimensional Brownian motion of a harmonic oscillator, if its mass is disregarded, where now $V( t)$ is interpreted as the coordinate of the oscillator, $-( m dV)/dt$ is the force of viscous friction, $- \beta V$ is a regular elastic force which forces the oscillator back to its equilibrium position, while $W ^ \prime ( t)$ is a random force which can be created by molecular shocks. In this way, the Ornstein–Uhlenbeck process also provides a model of the fluctuations for a harmonic oscillator performing a Brownian motion, analogous to the Einstein–Smoluchowski model of the Brownian motion of a free particle.

The Ornstein–Uhlenbeck process is a diffusion-type Markov process, homogeneous with respect to time (see Diffusion process); on the other hand, a process $V( t)$ which is at the same time a stationary random process, a Gaussian process and a Markov process, is necessarily an Ornstein–Uhlenbeck process. As a Markov process, the Ornstein–Uhlenbeck process can conveniently be characterized by its transition probability density $p( t, x, y)$, which is a fundamental solution of the corresponding Fokker–Planck equation (i.e. the forward Kolmogorov equation) of the form

$$\frac{\partial p }{\partial t } = \ \alpha \frac{\partial ( yp) }{\partial y } + \alpha \sigma ^ {2} \frac{\partial ^ {2} p }{\partial y ^ {2} } ,$$

and which, consequently, is given by the formula

$$p( t, x, y) = \frac{1}{[ 2 \pi \sigma ^ {2} ( 1- e ^ {- 2 \alpha t } )] ^ {2} } \mathop{\rm exp} \left \{ - \frac{( y- xe ^ {- \alpha t } ) ^ {2} }{2 \sigma ^ {2} ( 1- e ^ {- 2 \alpha t } ) } \right \} .$$

Many properties of the Ornstein–Uhlenbeck process $V( t)$( including its Markov property) can be deduced from known properties of a Wiener process, using the fact that the process

$$W _ {0} ( t) = \frac{\sqrt t } \sigma V \left ( \frac{ \mathop{\rm ln} t }{2 \alpha } \right )$$

is a standard Wiener process (see ). It therefore follows, in particular, that the realizations of an Ornstein–Uhlenbeck process are continuous and nowhere differentiable with probability 1, and that

$$\overline{\lim\limits}\; _ {t \rightarrow 0 } \frac{| V( t) - V( 0) | }{\sqrt {4 \alpha \sigma ^ {2} t \mathop{\rm ln} \mathop{\rm ln} \ {1/t } } } = 1,\ \ \overline{\lim\limits}\; _ {t \rightarrow \infty } \frac{| V( t) | }{\sqrt {2 \sigma ^ {2} \mathop{\rm ln} t } } = 1 ,$$

with probability 1.

How to Cite This Entry:
Ornstein–Uhlenbeck process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ornstein%E2%80%93Uhlenbeck_process&oldid=22866