  In this paper we develop a potential theory for 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 of the form \[ \Omega=\{(X,Y,t)=(x,x_{m},y,y_{m},t)\in\mathbb
R^{m-1}\times\mathbb R\times\mathbb R^{m-1}\times\mathbb R\times\mathbb R\mid
x_m>\psi(x,y,y_m,t)\}, \] where $\psi$ is assumed to satisfy a uniform
Lipschitz condition adapted to the dilation structure and the (non-Euclidean)
Lie group underlying the operator $\mathcal{L}$. Concerning $A=A(X,Y,t)$ we
assume that $A$ is bounded, measurable, symmetric and uniformly elliptic (as a
matrix in $\mathbb R^{m}$). Beyond the solvability of the Dirichlet problem and
other fundamental properties our results include scale and translation
invariant boundary comparison principles, boundary Harnack inequalities and
doubling properties of associated parabolic measures. All of our estimates are
translation- and scale-invariant with constants only depending on the constants
defining the boundedness and ellipticity of $A$ and the Lipschitz constant of
$\psi$. Our results represent a version, for operators of Kolmogorov type with
bounded, measurable coefficients, of the by now classical results of Fabes and
Safonov, any several others, concerning boundary estimates for uniformly
parabolic equations in (time-dependent) Lipschitz type domains.
