 Explicit Ornstein-Uhlenbeck solution

For deterministic $X_0=x$ and real $\lambda$, the Ornstein-Uhlenbeck stochastic differential equation $d{X}_t=-\lambda X_{t}dt+d{B}_t$ has solutionMultiply 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, isThus 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.