Exact Ornstein–Uhlenbeck momentum transition and covariance

Put $S=\sqrt{2\gamma {\beta }^{-1}}{M}^{1\mathrm{/}2}$. Multiplying the O equation by the deterministic integrating factor $e^{\gamma t}$ and applying [F3] componentwise gives $d(e^{\gamma t}{P}_t)=e^{\gamma t}Sd{W}_t$. Integrating from $0$ to $h$ and multiplying by $e^{-\gamma h}$ yields [F1, F2, F3, A1, given, algebra] $P_h=e^{-\gamma h}{P}_0+{\int }_0^h{e}^{-\gamma (h-s)}Sd{W}_s\mathrm{.}$ The integrand is deterministic and square-integrable. By [F2] the integral is centered Gaussian, independent of the starting sigma algebra, with covariance ${\int }_0^h{e}^{-2\gamma (h-s)}S{S}^{T}ds=2\gamma {\beta }^{-1}M{\int }_0^h{e}^{-2\gamma (h-s)}ds={\beta }^{-1}(1-e^{-2\gamma h})M\mathrm{.}$

By [F4], the displayed covariance has square root $\sqrt{{\beta }^{-1}(1-e^{-2\gamma h})}{M}^{1\mathrm{/}2}$. A centered Gaussian vector with this covariance has the same law as that matrix times $\xi \sim N_d(0,I_d)$; [F2] gives independence from the starting sigma algebra. This proves the exact transition and conditional moments.