  In this note we consider SDEs of the type $\mathrm{d} X_t=[F (X_t) -A X_t]
\mathrm{d} t +D \mathrm{d} W_t$ under the assumptions that $A$'s eigenvalues
are all of positive real parts and $F (\cdot)$ has slower-than-linear growth
rate. It is proved that $\displaystyle \varlimsup_{t \to \infty}
\frac{\|X_t\|}{\sqrt{\log t}} =\sqrt{2 \lambda_1}$ almost surely with
$\lambda_1$ being the largest eigenvalue of the matrix $\displaystyle \Sigma
:=\int_0^\infty e^{-s A} \cdot (D \cdot D^T) \cdot e^{-s A^T} \mathrm{d} s$;
the discarded measure-zero set can be chosen independent of the initial values
$X_0=x$.
