Colored noise and stage memory

For deterministic quadratic dynamics let $T(t,u)$ be its two-dimensional phase-space transition matrix, and let $e_p=(0,1{)}^{\top }$. The noise-induced displacement of $R=(x,p{)}^{\top }$ is

where the repeated letter $u$ in the argument of the stochastic process denotes time, not a second independent noise. Equivalently, writing the time variables as $a,b$, its covariance is