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Singularities of hypoelliptic Green functions. (English) Zbl 0909.60058
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. LĂ©andre, 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.

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)
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