A structure theorem for elliptic and parabolic operators with applications to homogenization of operators of Kolmogorov type

Abstract: 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 inequality 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.