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The Ornstein-Uhlenbeck Process: Stationary Distributions

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The Formal Theorem

Let {Xt}t≥0 \{X_t\}_{t \ge 0} be an Ornstein-Uhlenbeck process satisfying the stochastic differential equation dXt=θ(μ−Xt)dt+σdWt dX_t = \theta(\mu - X_t)dt + \sigma dW_t , where θ>0 \theta > 0 , μ∈R \mu \in \mathbb{R} , and σ>0 \sigma > 0 . As t→∞ t \to \infty , the transition probability density p(x,t∣x0,0) p(x, t | x_0, 0) converges to the stationary distribution π(x) \pi(x) , which is a normal distribution given by:
π(x)=θπσ2exp⁡(−θ(x−μ)2σ2) \pi(x) = \sqrt{\frac{\theta}{\pi \sigma^2}} \exp\left( -\frac{\theta(x - \mu)^2}{\sigma^2} \right)

Analytical Intuition.

Imagine a particle caught in a viscous fluid, tethered to a central 'anchor' μ \mu by an invisible, elastic rubber band. In a pure Wiener process, the particle wanders aimlessly, its variance ballooning to infinity. Here, however, the drift term θ(μ−Xt) \theta(\mu - X_t) acts as a restorative force: the further the particle drifts from μ \mu , the more aggressively it is pulled back. As the clock runs toward t→∞ t \to \infty , the system reaches a dynamic equilibrium. The kinetic energy injected by the random shocks σdWt \sigma dW_t is precisely counterbalanced by the damping friction θ \theta . We are no longer describing a particle that gets lost in the void, but one that has settled into a 'steady state'—a bell-shaped probability cloud centered firmly at μ \mu . The width of this cloud, determined by the ratio σ2/θ \sigma^2 / \theta , represents the perpetual tug-of-war between the chaotic noise trying to push the particle away and the rigid anchor pulling it home.
CAUTION

Institutional Warning.

Students frequently conflate the stationary variance with the time-dependent variance. Remember that while Var(Xt) \text{Var}(X_t) converges to σ2/2θ \sigma^2 / 2\theta as t→∞ t \to \infty , the transient variance is strictly σ22θ(1−e−2θt) \frac{\sigma^2}{2\theta}(1 - e^{-2\theta t}) , which only attains the stationary limit in the infinite time horizon.

Academic Inquiries.

01

Why is θ>0 \theta > 0 a strict requirement?

If θ≤0 \theta \le 0 , the restorative force becomes non-existent or repulsive, causing the variance of the process to diverge rather than converge, meaning no stationary distribution exists.

02

How does the Ornstein-Uhlenbeck process differ from Brownian motion?

Brownian motion has no central tendency or 'memory' of an equilibrium position, resulting in a variance that grows linearly with time, whereas the OU process is mean-reverting and maintains a stable long-term variance.

Standardized References.

  • Definitive Institutional SourceØksendal, B., Stochastic Differential Equations: An Introduction with Applications

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Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). The Ornstein-Uhlenbeck Process: Stationary Distributions: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/advanced-stochastic-processes/the-ornstein-uhlenbeck-process--stationary-distributions

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