Martingales, the Malliavin calculus and hypoellipticity under general Hörmander’s conditions. (English) Zbl 0445.60049

Bismut, Jean-Michel

Z. Wahrscheinlichkeitstheor. Verw. Geb. 56, 469-505 (1981).

MSC:

60H05	Stochastic integrals
60G44	Martingales with continuous parameter
60J65	Brownian motion

Keywords:

Malliavin calculus; hypoellipticity; Ornstein-Uhlenbeck operator; Girsanov transformation

Citations:

Zbl 0156.107; Zbl 0411.60060; Zbl 0427.60056

References:

[1]	Baxendale, P.: Wiener processes on Manifolds of maps. J. of Diff. Geometry, to appear · Zbl 0456.60074
[2]	Bismut, J.M.: Principes de mécanique aléatoire, to appear · Zbl 0528.60048
[3]	Bismut, J. M., Flots stochastiques et formule de Ito-Stratonovitch généralisée, CRAS, 290, 483-486 (1980) · Zbl 0428.60067
[4]	Bismut, J. M., A generalized formula of Ito and some other properties of stochastic flows, Z. Wahrscheinlichkeitstheorie verw. Gehiete, 55, 331-350 (1981) · Zbl 0456.60063
[5]	Bismut, J. M., An introductory approach to duality in optimal Stochastic control, SIAM Review, 20, 62-78 (1978) · Zbl 0378.93049
[6]	Clark, J. M.C., The representation of functionals of Brownian motion by stochastic integrals, Ann. Math. Stat., 41, 1282-1295 (1970) · Zbl 0213.19402
[7]	Dellacherie, C.; Meyer, P. A., Probabilités et Potentiels (1975), Paris: Hermann, Paris · Zbl 0323.60039
[8]	Elworthy, K. D.; Friedman, A.; Pinsky, M., Stochastic dynamical systems and their flows, Stochastic analysis, 79-95 (1978), New York: Acad. Press, New York · Zbl 0439.60065
[9]	Haussmann, U., Functionals of Ito processes as stochastic integrals, SIAM J. Control and Opt., 16, 252-269 (1978) · Zbl 0375.60070
[10]	Malliavin, P., Stochastic calculus of variations and hypoelliptic operators, 195-263 (1978), Tokyo: Kinokuniya, Tokyo · Zbl 0411.60060
[11]	Malliavin, P.; Friedman, A.; Pinsky, M., C_k-hypoellipticity with degeneracy, Stochastic Analysis, 199-214 (1978), New York and London: Acad. Press, New York and London · Zbl 0449.58022
[12]	Stroock, D.: The Malliavin calculus and its application to second order parabolic differential equations, Preprint 1980
[13]	Stroock, D. W.; Varadhan, S. R.S., Multidimensional diffusion processes, Grundlehren der Mathematischen Wissenschaften (1979), Berlin-Heidelberg-New York: Springer, Berlin-Heidelberg-New York · Zbl 0426.60069
[14]	Jacod, J.; Yor, M., Etude des solutions extrémales et représentation intégrale des solutions pour certains problèmes de martingales, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 38, 83-125 (1977) · Zbl 0346.60032
[15]	Hörmander, L., Hypoelliptic second order differential equations, Acta Math., 119, 147-171 (1967) · Zbl 0156.10701
[16]	Hörmander, L.; Melin, A., Free systems of vector fields, Ark. Mat., 16, 83-88 (1978) · Zbl 0383.35013
[17]	Rothschild, L. P.; Stein, E. M., Hypoelliptic differential operators and nilpotent groups, Acta Math., 137, 247-320 (1976) · Zbl 0346.35030
[18]	Abraham, R.; Marsden, J., Foundations of mechanics (1978), Reading: Benjamin, Reading · Zbl 0393.70001
[19]	Haussmann, U., On the integral representation of functionals of Ito processes, Stochastic, 3, 17-27 (1979) · Zbl 0427.60056
[20]	Ichihara, K.; Kunita, H., A classification of second order degenerate elliptic operators and its probabilistic characterization, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 30, 235-254 (1974) · Zbl 0326.60097
[21]	Davis, M. H.A., Functionals of diffusion processes as stochastic integrals, Math. Proc. Cambridge Philos. Soc., 87, 157-166 (1980) · Zbl 0424.60063
[22]	Ikeda, N., Watanabe, S.: Diffusions on manifolds. To appear · Zbl 0264.60052
[23]	Chevalley, C., Theory of Lie groups. Vol. I (1946), Princeton: Princeton University Press, Princeton · Zbl 0063.00842