  In this paper we consider the symmetric Kolmogorov operator $L=\Delta
+\frac{\nabla \mu}{\mu}\cdot \nabla$ on $L^2(\mathbb R^N,d\mu)$, where $\mu$ is
the density of a probability measure on $\mathbb R^N$. Under general conditions
on $\mu$ we prove first weighted Rellich's inequalities with optimal constants
and deduce that the operators $L$ and $-L^2$ with domain $H^2(\mathbb
R^N,d\mu)$ and $H^4(\mathbb R^N,d\mu)$ respectively, generate analytic
semigroups of contractions on $L^2(\mathbb R^N,d\mu)$. We observe that $d\mu$
is the unique invariant measure for the semigroup generated by $-L^2$ and as a
consequence we describe the asymptotic behaviour of such semigroup and obtain
some local positivity properties. As an application we study the
bi-Ornstein-Uhlenbeck operator and its semigroup on $L^2(\mathbb R^N,d\mu)$.
