  We consider properties of second-order operators $H = -\sum^d_{i,j=1}
\partial_i \, c_{ij} \, \partial_j$ on $\Ri^d$ with bounded real symmetric
measurable coefficients. We assume that $C = (c_{ij}) \geq 0$ almost
everywhere, but allow for the possibility that $C$ is singular. We associate
with $H$ a canonical self-adjoint viscosity operator $H_0$ and examine
properties of the viscosity semigroup $S^{(0)}$ generated by $H_0$. The
semigroup extends to a positive contraction semigroup on the $L_p$-spaces with
$p \in [1,\infty]$. We establish that it conserves probability, satisfies
$L_2$~off-diagonal bounds and that the wave equation associated with $H_0$ has
finite speed of propagation. Nevertheless $S^{(0)}$ is not always strictly
positive because separation of the system can occur even for subelliptic
operators. This demonstrates that subelliptic semigroups are not ergodic in
general and their kernels are neither strictly positive nor H\"older
continuous. In particular one can construct examples for which both upper and
lower Gaussian bounds fail even with coefficients in $C^{2-\varepsilon}(\Ri^d)$
with $\varepsilon > 0$.
