Boundary Harnack inequalities for Kolmogorov equations in asymptotically cylindrical Lipschitz domains

Abstract

We establish boundary Harnack inequalities for non-negative solutions of the Kolmogorov equation \[ \mathcal Ku=\Delta_xu+x\cdot\nabla_yu-\partial_tu=0 \] in non-characteristic intrinsic Lipschitz graph domains whose defining functions may depend on the higher-order variable \(y_m\). Earlier theory required independence of \(y_m\). We replace this symmetry by quantitative asymptotic cylindricality, measured by a scale-invariant \(C^{0,1/3}_{y_m}\) defect. The perturbative comparison theorem applies whenever this defect is sufficiently small, and hence at all sufficiently small scales for \(C^{0,\alpha}_{y_m}\) graphs with \(\alpha>1/3\), as well as under a vanishing critical cylindrical modulus. We obtain local comparability and intrinsic H\"older continuity of quotients of positive solutions, together with a H\"older estimate for their logarithmic quotient. The proof combines local boundary estimates, valid without \(y_m\)-independence, with a global rigidity argument for cylindrical blow-up limits. The central issue is uniform balance of forward and backward reference values. If this balance fails at collapsing scales, maximal-scale selection and a second rescaling produce a non-negative polynomial-growth Dirichlet solution in an unbounded cylindrical domain that vanishes at the backward reference point but not at the forward one. We exclude this configuration using boundary-energy and mean-value estimates, finite-dimensionality of spaces of polynomial-growth solutions, and translation invariance in \(y_m\). The resulting finite-dimensional translation representation yields real-analytic dependence on \(y_m\). Propagation of zeros, analytic continuation along invariant fibres, and a Tikhonov uniqueness argument then force the limiting solution to vanish identically. The exponent \(1/3\) is critical because \(y_m\) has homogeneous degree three.