Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 202 / 6 / f / Solution

If $X_0\sim N(0,(2\lambda {)}^{-1})$ independently of $B$, thenThus $X_t\sim N(0,(2\lambda {)}^{-1})$ for every $t$. For $0<s<t$, the Markov decompositionhas an increment independent of $X_s$, and henceThis is the stationary Ornstein-Uhlenbeck covariance.