  We consider the operators \[
  \nabla_X\cdot(A(X)\nabla_X),\ \nabla_X\cdot(A(X)\nabla_X)-\partial_t,\
\nabla_X\cdot(A(X)\nabla_X)+X\cdot\nabla_Y-\partial_t, \] where $X\in \Omega$,
$(X,t)\in \Omega\times \mathbb R$ and $(X,Y,t)\in \Omega\times \mathbb
R^m\times \mathbb R$, respectively, and where $\Omega\subset\mathbb R^m$ is a
(unbounded) Lipschitz domain with defining function $\psi:\mathbb
R^{m-1}\to\mathbb R$ being Lipschitz with constant bounded by $M$. Assume that
the elliptic measure associated to the first of these operators is mutually
absolutely continuous with respect to the surface measure $\mathrm{d}
\sigma(X)$, and that the corresponding Radon-Nikodym derivative or Poisson
kernel satisfies a scale invariant reverse H\"older inequalities in $L^p$, for
some fixed $p$, $1<p<\infty$, with constants depending only on the constants of
$A$, $m$ and the Lipschitz constant of $\psi$, $M$. Under this assumption we
prove that then the same conclusions are also true for the parabolic measures
associated to the second and third operator with $\mathrm{d} \sigma(X)$
replaced by the surface measures $\mathrm{d} \sigma(X)\mathrm{d} t$ and
$\mathrm{d} \sigma(X)\mathrm{d} Y\mathrm{d} t$, respectively. This structural
theorem allows us to reprove several results previously established in the
literature as well as to deduce new results in, for example, the context of
homogenization for operators of Kolmogorov type. Our proof of the structural
theorem is based on recent results established by the authors concerning
boundary Harnack inequalities for operators of Kolmogorov type in divergence
form with bounded, measurable and uniformly elliptic coefficients.
