  We analyze degenerate, second-order, elliptic operators $H$ in divergence
form on $L_2({\bf R}^{n}\times{\bf R}^{m})$. We assume the coefficients are
real symmetric and $a_1H_\delta\geq H\geq a_2H_\delta$ for some $a_1,a_2>0$
where \[ H_\delta=-\nabla_{x_1} c_{\delta_1, \delta'_1}(x_1)
\nabla_{x_1}-c_{\delta_2, \delta'_2}(x_1) \nabla_{x_2}^2 . \] Here $x_1\in{\bf
R}^n$, $x_2\in{\bf R}^m$ and $c_{\delta_i, \delta'_i}$ are positive measurable
functions such that $c_{\delta_i, \delta'_i}(x)$ behaves like $|x|^{\delta_i}$
as $x\to0$ and $|x|^{\delta_i'}$ as $x\to\infty$ with
$\delta_1,\delta_1'\in[0,1>$ and $\delta_2,\delta_2'\geq0$.
  Our principal results state that the submarkovian semigroup $S_t=e^{-tH}$ is
conservative and its kernel $K_t$ satisfies bounds \[ 0\leq K_t(x ;y)\leq a
(|B(x ;t^{1/2})| |B(y ;t^{1/2})|)^{-1/2} \] where $|B(x ;r)|$ denotes the
volume of the ball $B(x ;r)$ centred at $x$ with radius $r$ measured with
respect to the Riemannian distance associated with $H$. The proofs depend on
detailed subelliptic estimations on $H$, a precise characterization of the
Riemannian distance and the corresponding volumes and wave equation techniques
which exploit the finite speed of propagation.
  We discuss further implications of these bounds and give explicit examples
that show the kernel is not necessarily strictly positive, nor continuous.
