Abstract
This paper is devoted to a precise description of the singularity near the diagonal of the Green function associated to a hypoelliptic operator using a probabilistic approach. Examples and some applications to potential theory are given.
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Ben Arous, G.: ‘Flots et séries de Taylor stochastiques’, Probab. Th. Rel. Fields 81 (1989), 29–77.
Ben Arous, G.: ‘Développement asymptotique du noyau de la chaleur hypoelliptique sur la diagonale’, Ann. Inst. Fourier 39 (1989), 73–99.
Ben Arous, G. and Léandre, R.: ‘Décroissance exponentielle du noyau de la chaleur sur la diagonale I, II’, Probab. Th. Rel. Fields 90 (1991), 175–202, 377–402.
Beals, R., Gaveau, B. and Greiner, P. C.: ‘The Green Function of Model Step Two Hypoelliptic Operators and the Analysis of Certain Tangential Cauchy Riemann Complexes’, Adv. Math. 121 (1996), 288–345.
Biroli, M.: ‘The Wiener Test for Poincaré-Dirichlet Forms’, in: GowriSankaran K., et al. (eds.), Classical and Modern Potential Theory and Applications, Kluwer Academic Publishers, Dordrecht, Boston, London, 1994, 93–104.
Blumenthal, R. M. and Getoor, R. K.: Markov processes and potential theory, Academic Press, New York, London, 1968.
Bony, J. M.: ‘Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés’, Ann. Inst. Fourier 19 (1969), 277–304.
Castell, F.: ‘Asymptotic expansion of stochastic flows’, Probab. Th. Rel. Fields 96 (1993), 225–239.
Chaleyat-Maurel, M. and Le Gall, J.-F.: ‘Green function, capacity and sample paths properties for a class of hypoelliptic diffusions processes’, Probab, Th. Rel. Fields 83 (1989), 219–264.
Fefferman, C. L. and Sánchez-Calle, A.: ‘Fundamental solutions for second order subelliptic operators’, Ann. Math. 124 (1986), 247–272.
Folland, G. B.: ‘A fundamental solution for a subelliptic operator’, Bull. Amer. Math. Soc. 79 (1973), 373–376.
Gallardo, L.: ‘Capacités, mouvement brownien et problème de l'épine de Lebesgue sur les groupes de Lie nilpotents’, in: Heyer, H. (ed.), Probability measures on groups, Proceedings of the Conference at Oberwolfach 1981, Lecture Notes Math. 928, Springer-Verlag, Berlin, Heidelberg, New York, 1982, 96–120.
Gaveau, B.: ‘Principe de moindre action, propagation de la chaleur et estimées sous-elliptiques sur certains groupes nilpotents’, Acta Math. 139 (1977), 96–153.
Gradinaru, M.: Fonctions de Green et support de diffusions hypoelliptiques, Thèse, Université de Paris-Sud, Orsay, 1995.
Greiner, P. C.: ‘A fundamental solution for a nonelliptic partial differential operator’, Canad. Jour. Math. 31 (1979), 1107–1120.
Greiner, P. C.: ‘On the second order hypoelliptic differential operators and the ∂-Neumann problem’, in: Diedrich, K. (ed.), Complex analysis, Proceedings of Workshop at Wuppertal 1990, Vieweg, Braunschweig, 1991, 134–142.
Greiner, P. C. and Stein, E. M.: ‘On the solvability of some differential operators of type □□’, in: Kohn, J. and Vesentini, E. (eds.), Several complex variables, Proceedings of the conference at Cortona 1976–1977, Scuola Normale Superiore, Pisa, 1978, 106–165.
Léandre, R.: ‘Développement asymptotique de la densité d'une diffusion dégénéré’, Forum Math. 4 (1992), 45–75.
Nagel, A., Stein, E. M. and Wainger, S.: ‘Balls and metrics defined by vector fields I. Basic properties’, Acta Math. 155 (1985), 103–147.
Sánchez-Calle, A.: ‘Fundamental solutions and geometry of the sum of square of vector fields’, Invent. Math. 78 (1984), 143–160.
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Arous, G.B., Gradinaru, M. Singularities of Hypoelliptic Green Functions. Potential Analysis 8, 217–258 (1998). https://doi.org/10.1023/A:1008608825872
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DOI: https://doi.org/10.1023/A:1008608825872