= Explicit Ornstein-Uhlenbeck solution
For deterministic $X_0=x$ and real $\lambda$, the <Ornstein-Uhlenbeck stochastic differential equation> $dX_t=-\lambda X_t\,dt+dB_t$ has solution
$$
X_t=xe^{-\lambda t}+\int_0^te^{-\lambda(t-s)}\,dB_s.
$$
Multiply by the integrating factor $e^{\lambda t}$ and apply the <Itô product rule> to obtain it. The <stochastic integral> is centered Gaussian and its <variance>, by the <Itô isometry>, is
$$
\operatorname{Var}(X_t)=\int_0^te^{-2\lambda(t-s)}ds=
\begin{cases}(1-e^{-2\lambda t})/(2\lambda),&\lambda\ne0,\\t,&\lambda=0.\end{cases}
$$
Thus the apparent zero-denominator case is ordinary <Brownian motion>, and negative $\lambda$ gives growing rather than mean-reverting fluctuations.
For $\lambda>0$, take an <independent> initial value $X_0\sim N(0,(2\lambda)^{-1})$. Adding its <variance> contribution gives the time-independent <variance> $(2\lambda)^{-1}$ and <covariance function> $e^{-\lambda|t-s|}/(2\lambda)$, making the process stationary. A deterministic zero start instead has the transient <variance> in the display. These are different initial ensembles, so their <variances> should not be interchanged.
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