Ben Arous, Gérard; Kusuoka, Shigeo; Stroock, Daniel W. The Poisson kernel for certain degenerate elliptic operators. (English) Zbl 0556.35036 J. Funct. Anal. 56, 171-209 (1984). Let \(V_ 0,V_ 1,...,V_ d\) be given vector fields and consider the differential operator \(L=\sum^{d}_{k=1}V^ 2_ k+V_ 0\) where the vector fields are interpreted as directional derivatives, and let \(G\) be a bounded open set with smooth non-characteristic boundary. If the Dirichlet problem for \(L\) on \(G\) is well posed then there exists a weakly continuous map \(x\to H(x,\cdot)\) from \(G\) to probability measures on \(\partial G\) such that \(u_ f(x)=\int f(y)H(x,dy),\) \(x\in G\), \(f\in C(\partial G)\) is the solution of the Dirichlet problem, \(Lu_ f=0\) in \(G\), \(u_ f=f\) on \(\partial G\). The present article studies the harmonic measure \(H(x,\cdot)\). It is shown that \(H(x,\cdot)\) admits a smooth density \(h(x,\cdot)\) with respect to the natural boundary measure, provided that certain Hörmander-type nondegeneracy conditions and strong nontangency conditions at the boundary are satisfied. The approach differs from the usual analytic strategy and can be viewed as a generalization of the classical subordination procedure used to study the Poisson kernel in the upper half space. Reviewer: G. Gudmundsdottir Cited in 1 ReviewCited in 9 Documents MSC: 35J25 Boundary value problems for second-order elliptic equations 35J70 Degenerate elliptic equations Keywords:non-characteristic boundary; Dirichlet problem; harmonic measure; Hörmander-type nondegeneracy; subordination procedure; Poisson kernel PDFBibTeX XMLCite \textit{G. Ben Arous} et al., J. Funct. Anal. 56, 171--209 (1984; Zbl 0556.35036) Full Text: DOI References: [1] Bismut-Michel, Diffusions conditionnelles, J. Funct. Anal., 44, 174-211 (1981) · Zbl 0475.60061 [2] Derridj, M., Un problème aux limites pour une classe d’operateurs hypoelliptiques du second ordre, Ann. Inst. Fourier (Grenoble), 21, 4, 99-148 (1971) · Zbl 0215.45405 [3] Hörmander, L., Hypoelliptic second order differential equations, Acta Math., 119, 147-171 (1967) · Zbl 0156.10701 [4] Jerison, D., The Dirichlet problem for the Kohn Laplacian on the Heisenberg group, II, J. Funct. Anal., 43, 2, 224-257 (1981) · Zbl 0493.58022 [6] Stroock, D.; Varadhan, S. R.S, On degenerate elliptic-parabolic operators of second order and their associated diffusions, Comm. Pure Appl. Math., 25, 651-714 (1972) · Zbl 0344.35041 [7] Stroock, D.; Varadhan, S. R.S, Multidimensional Diffusion Processes, (Grundlehren 233 (1979), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York) · Zbl 1316.60124 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.