  We consider strongly degenerate parabolic operators of the form \[
  \mathcal{L}:=\nabla_X\cdot(A(X,Y,t)\nabla_X)+X\cdot\nabla_Y-\partial_t \] in
unbounded domains \[
  \Omega=\{(X,Y,t)=(x,x_{m},y,y_{m},t)\in\mathbb R^{m-1}\times\mathbb
R\times\mathbb R^{m-1}\times\mathbb R\times\mathbb R\mid x_m>\psi(x,y,t)\}. \]
We assume that $A=A(X,Y,t)$ is bounded, measurable and uniformly elliptic (as a
matrix in $\mathbb R^{m}$) and concerning $\psi$ and $\Omega$ we assume that
$\Omega$ is what we call an (unbounded) Lipschitz domain: $\psi$ satisfies a
uniform Lipschitz condition adapted to the dilation structure and the
(non-Euclidean) Lie group underlying the operator $\mathcal{L}$. We prove,
assuming in addition that $\psi$ is independent of the variable $y_m$, that
$\psi$ satisfies an additional regularity condition formulated in terms of a
Carleson measure, and additional conditions on $A$, that the associated
parabolic measure is absolutely continuous with respect to a surface measure
and that the associated Radon-Nikodym derivative defines an $A_\infty$-weight
with respect to the surface measure.
