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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.

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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