Singularities of hypoelliptic Green functions. (Singularités des fonctions de Green hypoelliptiques.) (French) Zbl 0868.58083

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.


58J65 Diffusion processes and stochastic analysis on manifolds
31C99 Generalizations of potential theory
Full Text: DOI Numdam EuDML


[1] Azencott, R., : Formule de Taylor stochastique et développements asymptotiques d’intégrales de Feynmann, Dans: Azéma, J., Yor, M.Seminaire de Probabilités XVI. Supplément: Géométrie différentielle stochastique ( vol. 921, pp. 237-284), Berlin Heidelberg , New York: Springer1982 · Zbl 0484.60064
[2] Ben Arous, G.: Flots et séries de Taylor stochastiques, Probab. Th. Rel. Fields81, pp. 29-77 (1989) · Zbl 0639.60062
[3] Ben Arous, G.: Développement asymptotique du noyau de la chaleur hypoelliptique sur la diagonale, Ann. Inst. Fourier39, pp. 73-99 (1989) · Zbl 0659.35024
[4] Ben Arous, G., Gradinaru, M.: Singularities of hypoelliptic Green functions, Preprint LMENS-95-19, soumis au “Potential Analysis”, 1995 · Zbl 0909.60058
[5] Ben Arous, G., Léandre, R.: Décroissance exponentielle du noyau de la chaleur sur la diagonale I,II, Probab. Th. Rel. Fields90, pp. 175-202377-402, (1991) · Zbl 0734.60026
[6] Beals, R., Gaveau, B., Greiner, P.C.: Solution fondamentale pour des varietés de Cauchy-Riemann, Exposés au Seminaire d’Analyse, Institut “Henri Poincaré”, 1993
[7] Castell, F.: Asymptotic expansion of stochastic flows, Probab. Th. Rel. Fields96, pp. 225-239 (1993) · Zbl 0794.60054
[8] Chaleyat-Maurel, M., Le Gall, J.-F.: Green function, capacity and sample paths properties for a class of hypoelliptic diffusions processes, Probab. Th. Rel. Fields83, pp. 219-264 (1989) · Zbl 0686.60058
[9] Folland, G.B.: A fundamental solution for a subelliptic operator, Bull. Amer. Math. Soc.79, pp. 373-376 (1973) · Zbl 0256.35020
[10] Folland, G.B., Stein, E.M.: Estimates for the ∂b-complex and analysis on the Heisenberg group, Comm. Pure Appl. Math.27, pp. 429-522 (1974) · Zbl 0293.35012
[11] Fefferman, C.L., Sánchez-Calle, A.: Fundamental solutions for second order subelliptic operators, Ann. Math.124, pp. 247-272 (1986) · Zbl 0613.35002
[12] Gradinaru, M., : Fonctions de Green et support de diffusions hypoelliptiques, Thèse, Université de Paris-Sud, Orsay, 1995
[13] Gaveau, B.: Principe de moindre action, propagation de la chaleur et estimées sous-elliptiques sur certains groupes nilpotents, Acta Math.139, pp. 96-153 (1977) · Zbl 0366.22010
[14] Greiner, P.C.: A fundamental solution for a nonelliptic partial differential operator, Canad. Jour. Math.31, pp. 1107-1120 (1979) · Zbl 0475.35003
[15] Greiner, P.C.: On second order hypoelliptic differential operators and the ∂-Neumann problem, In: Diedrich, K, pp. 134-142, Braunschweig: Vieweg1991 · Zbl 0747.58045
[16] Greiner, P.C., Stein, E.M.: On the solvability of some differential operators of type □b, Dans: Several complex variables, Proceedings of the conference at Cortona 1976-1977, pp. 106-165, Pisa: Scuola Normale Superiore1978 · Zbl 0434.35007
[17] Hörmander, L., : Hypoelliptic second order differential equations, Acta Math.119, pp. 147-171 (1967) · Zbl 0156.10701
[18] Jerison, D., Sánchez-Calle, A.: Subelliptic second order differential operators, Dans: Berenstein, C.A (ed.)Complex analysis III, Proceedings of the Special Year at University of Maryland1985-1986, ( vol. 1277 pp. 46-77), , BerlinHeidelberg1987SpringerNew York · Zbl 0634.35017
[19] Léandre, R.: Développement asymptotique de la densité d’une diffusion dégénérée, Forum Math.4, pp. 45-75 (1992) · Zbl 0749.60054
[20] Nagel, A., Stein, E.M., Wainger, S.: Balls and metrics defined by vector fields I Basic properties. Acta Math.155, pp. 103-147 (1985) · Zbl 0578.32044
[21] Sánchez-Calle, A.: Fundamental solutions and geometry of the sum of square of vector fields, Invent. math.78, pp. 143-160 (1984) · Zbl 0582.58004
[22] Sznitman, A.S.: Some bounds and limiting results for the measure of the Wiener sausage of small radius associated with elliptic diffusions, Stoch. Proc. Appl.25, pp. 1-25 (1987). · Zbl 0628.60080
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.