= Ornstein-Uhlenbeck stationary initialization
{c}
{title2=$X_0\sim\mathcal N(\mu,\sigma^2/(2\theta))$}
Set $v_\infty=\sigma^2/(2\theta)$ and draw $X_0\sim\mathcal N(\mu,v_\infty)$ independently of future <Brownian increments>. The <Ornstein-Uhlenbeck process> is then a <strictly stationary process>, with
$$
\mathbb E X_t=\mu,\qquad \operatorname{Var}(X_t)=v_\infty,\qquad \operatorname{Cov}(X_s,X_t)=v_\infty e^{-\theta|t-s|}.
$$
Every time has the same <normal distribution>, and the <covariance function> depends only on the time lag. For a <Gaussian process> these properties also make all its <finite-dimensional distributions> invariant under a common time shift. The <stationary distribution> balances the continual random forcing against the contracting <drift>.<Applied Stochastic Differential Equations by Simo Särkkä and Arno Solin>[Applied Stochastic Differential Equations, Example 6.8, printed pp. 91–92, equations (6.73)–(6.76): steady-state variance and covariance]{ref}
With a fixed $X_0=x_0$, the <covariance function> instead is
$$
\operatorname{Cov}(X_s,X_t)=v_\infty\left(e^{-\theta|t-s|}-e^{-\theta(s+t)}\right),\qquad s,t\ge0.
$$
In particular, it is zero when $s=0$, since the initial value has no uncertainty. The second term records the fixed start; it is the general scaling of the <zero-start Ornstein-Uhlenbeck covariance>. The pure <exponential covariance function> displayed above assumes stationary initialization. For a fixed start, that function emerges as the late-time limit at fixed lag, while the one-time <probability distributions> approach $\mathcal N(\mu,v_\infty)$ by <convergence in distribution>.
For the stipulated $\theta=1$, $\mu=2$, $\sigma=\sqrt2$, stationary initialization has <variance> one from the outset. Its <autocorrelation> is $e^{-|h|}$, so the <correlation time> is one: a lag of one reduces the <autocorrelation> to $e^{-1}\approx0.3679$. Its raw second moment $\mathbb E[X_sX_t]=4+e^{-|t-s|}$ includes the squared <expected value>; it must not be confused with the <covariance>.
\Image[/ornstein-uhlenbeck-stationary-covariance-samples.png]
{title=Simulated stationary Ornstein-Uhlenbeck samples and analytic covariance compared with a fixed start}
The upper panel shows three simulated sequences with independent stationary initial values and exact transitions at spacing $0.05$. Joining their points only guides the eye; it does not simulate a continuous <sample path> between those points. The lower panel shows analytic <covariance functions>: the stationary curve and, for comparison, $\operatorname{Cov}(X_s,X_{s+h})$ for fixed initialization at two different ages $s$. These are simulated samples and model curves, not measured trajectories or empirical <covariance> estimates.
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