Ben Arous, Gerard; Gradinaru, Mihai Singularities of hypoelliptic Green functions. (English) Zbl 0909.60058 Potential Anal. 8, No. 3, 217-258 (1998). The paper is devoted to a precise description of the singularity near the diagonal of the Green function associated to a hypoelliptic operator. The behaviour is in contrast with the elliptic situation, the Heisenberg group situation or the “curved” Heisenberg group situation studied by M. Chaleyat-Maurel and J.-F. Le Gall [Probab. Theory Relat. Fields 83, No. 1, 219-264 (1989; Zbl 0686.60058)]. Also, the result can be compared with the results on heat kernel [see G. Ben Arous, Ann. Inst. Fourier 39, No. 1, 73-99 (1989; Zbl 0659.35024) and R. Léandre, Forum Math. 4, No. 1, 45-75 (1992; Zbl 0749.60054)]. The approach is probabilistic. It relies on results on stochastic Taylor expansion of paths of the degenerate diffusion associated to the operator [see G. Ben Arous, Probab. Theory Relat. Fields 81, No. 1, 29-77 (1989; Zbl 0639.60062) and F. Castell, ibid. 96, No. 2, 225-239 (1993; Zbl 0794.60054)], and also on the a priori estimate given by A. Nagel, E. M. Stein and S. Wainger [Acta Math. 155, 103-147 (1985; Zbl 0578.32044)]. Examples with explicit computations and applications to potential theory (as estimates of the capacities of small sets) are given. Reviewer: Mihai Gradinaru (Nancy) Cited in 1 ReviewCited in 3 Documents MSC: 60J60 Diffusion processes 60J45 Probabilistic potential theory 65H10 Numerical computation of solutions to systems of equations 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) Keywords:degenerate diffusion; hypoelliptic operator; Green function; Taylor stochastic expansion; capacity Citations:Zbl 0686.60058; Zbl 0659.35024; Zbl 0749.60054; Zbl 0639.60062; Zbl 0794.60054; Zbl 0578.32044 PDFBibTeX XMLCite \textit{G. Ben Arous} and \textit{M. Gradinaru}, Potential Anal. 8, No. 3, 217--258 (1998; Zbl 0909.60058) Full Text: DOI