  We consider the inverse problem of determining initial data in general
Ornstein-Uhlenbeck equations on the Euclidean space from partial measurement
localized on the so-called thick sets. Using the logarithmic convexity
technique and recent observability results, we prove new stability estimates of
logarithmic rate for large classes of initial data. Such stability estimates
are crucial when dealing with the numerical reconstruction of initial data. Our
analysis covers both cases: the analytic Ornstein-Uhlenbeck semigroup on
$L^2\left(\mathbb R^N, \mathrm{d}\mu\right)$ with invariant measure $\mu$, and
the non-analytic Ornstein-Uhlenbeck semigroup on $L^2\left(\mathbb R^N,
\mathrm{d} x\right)$ with the Lebesgue measure. We treat general equations with
a diffusion matrix and the Ornstein-Uhlenbeck equation with fractional
diffusion in the latter case. This allows us to extend some recent results and
simplify some parts of the proof.
