= Ornstein-Uhlenbeck process
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{wiki=Ornstein–Uhlenbeck_process}
= OU process
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An Ornstein-Uhlenbeck process combines <mean reversion> with continuing random fluctuations. In the scalar model, its <stochastic differential equation> is
$$
dX_t=-\theta(X_t-\mu)\,dt+\sigma\,dW_t,\qquad \theta>0,\quad \sigma>0,\quad \mu\in\mathbb R,
$$
where $W$ is a standard <Brownian motion>. The <drift> points towards $\mu$: it is negative above $\mu$ and positive below it. The coefficient $\theta$ sets the speed of <mean reversion>, while $\sigma$ sets the strength of the random forcing. We consider a fixed or <normal> initial value independent of future <Brownian increments>, giving a <Gaussian process> and a <Markov process>.
For a stipulated example, use $\theta=1$, $\mu=2$, $\sigma=\sqrt2$ and the fixed initial value $X_0=6$, in arbitrary consistent state and time units. Then the <expected value> relaxes as $2+4e^{-t}$ and the <variance> grows as $1-e^{-2t}$. At time one they are about $3.4715$ and $0.8647$; the limiting <variance> is one. The <sample paths> keep fluctuating even after the <expected value> is close to two. The <Ornstein-Uhlenbeck transition distribution> describes this evolving uncertainty; <Ornstein-Uhlenbeck stationary initialization> instead starts with that limiting <normal distribution> already in place.
A physical motivation is the <velocity> of a particle undergoing <Brownian motion>: <linear drag> removes a persistent initial <velocity>, while random forcing continually renews it. The original calculation gives a <normal distribution> for the <velocity> after subtracting its exponentially decaying initial contribution, with decay rate equal to the friction coefficient divided by the particle's <mass>.<On the Theory of the Brownian Motion by G. E. Uhlenbeck and L. S. Ornstein>[On the Theory of the Brownian Motion, abstract: velocity minus its decaying initial contribution; friction coefficient divided by mass]{ref} A position coordinate in an overdamped <harmonic optical trap> gives another interpretation of the same scalar equation, with $\theta=k/\zeta$ for <spring constant> $k$ and drag coefficient $\zeta$. Its physical assumptions and connection to measurements are developed in <thermal calibration of an optical trap>.
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