Solutions positives et mesure harmonique pour des opérateurs paraboliques dans des ouverts «lipschitziens»

Abstract

Let L be a parabolic operator on R n+1 written in divergence form and with Lipschitz coefficients relatively to an adapted metric. We compare, near the boundary, the relative behavior of positive L-solutions on a Lipschitz domain. We first establish a so-called weak boundary Harnack principle. We then establish a uniform Harnack principle for certain particular positive L-solutions. This principle then allows us to prove another strong boundary Harnack principle for certain pairs of positive L-solutions. Then, we can generalize to L-operators some of J. T. Kemper results: we characterize the Martin boundary for “Lipschitz” domains and we show that the positive L-solutions on such domains admit non tangential limits except for a negligible set with respect to harmonic measure. Finally, in the last part, and for slightly more regular domains, we establish the equivalence between harmonic measure, adjoint harmonic measure and surface measure thus developing some results of J. M. Wu and R. Kaufman.