Abstract
We give an adaptation of the Malliavin calculus of Bismut type for a semi-group generated by a four order operator on a Lie group. In such a case, the semi-group does not preserve the positivity.
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Albeverio, S., Hoegh-Krohn, R.: Dirichlet forms and diffusion processes on rigged Hilbert spaces. Z. Wahr. Verw. Geb. 40, 1–57 (1977)
Berezanskii, Y.: The selfadjointness of elliptic operators with infnite number of variables. Ukr. J. Math. 27, 729–742 (1975)
Bismut, J.M.: Martingales, the Malliavin calculus and hypoellipticity under general Hoermander’s conditions. Z. Wahr. Verw. Geb. 63, 469–505 (1981)
Chazarain, J., Piriou, A.: Introduction a la théorie des équations aux dérivées partielles linéaires. Gauthier-Villars, Paris (1981)
Davies, E.B.: \(L^p\) spectral theory of higher-order elliptic differential equations. Bull. Lond. Math. Soc. 29, 513–546 (1997)
Dieudonné, J.: Eléments d’analyse, vol. VII. Gauthiers-Villars, Paris (1977)
Gross, L.: Potential theory on a Hilbert space. J. Funct. Anal. 1, 123–181 (1966)
Hida, T.: Analysis of Brownian Functionals. Carleton Mathematical Lecture Notes, vol. 13. Carleton University Press, Ottawa (1975)
Hoermander, L.: The Analysis of Linear Partial Operators IV. Springer, Berlin (1984)
Hoermander, L.: The Analysis of Linear Partial Operators III. Springer, Berlin (1984)
Léandre, R.: A Girsanov formula associated to a big order pseudodifferential operator. In: Festchrift in honour of G. N’Guérékata, B. Anglade et al. (eds.) Cubo: a Mathematical Journal, vol. 15, pp. 113–119 (2013)
Léandre, R.: Applications of the Malliavin calculus of Bismut type without probability”. In: Madureira, A.M., W.S.E.A.S., CD. (eds.) 6th International Conference on Simulation, Modeling and Optimization, WSEAS Transactions on Mathematics, vol. 5, pp. 559–564, 1205–1210 (2006)
Léandre, R.: Applications quantitatives et qualitatives du Calcul de Malliavin. In: Séminaire Franco-Japonais, S., Métivier, M., Watanabe, S. (eds.) Lectures Notes Mathematics, pp. 109–134. Springer, Berlin (1988). English translation: In: Geometry of Random Motion. Durrett, R., Pinsky, M. (eds.) Contemporary Mathematics 73, pp. 173–196. A.M.S., Providence (1988)
Léandre, R.: Large deviation estimates for a non-Markovian Lévy generator of big order. In: Vagenas, E., et al. (eds.) 4th International Conference on Mathematical Modern Physical Sciences. Journal of Physics: Conference Series, vol. 633, pp. 012085 (2015)
Léandre, R.: Malliavin of Bismut type without probability. In: Festchrift in Honour of K. Sinha, V.S. Sunder (eds.) Proceedings of the Indian Academy of Science and Mathematical Science, vol. 116, pp. 507–518 (2006)
Léandre, R.: Perturbation of the Malliavin calculus of Bismut type of large order. In: Gazeau, J.P., et al. (eds.) To appear in XXXI International Congress Group of Methods in Physics
Léandre, R.: The Itô–Stratonovitch formula for an operator of order four. In: Festchrift in honour of A. Mikherjea, P. Feinsilver, S. Mohammed. Budzhan, G., et al. (eds.) Contemporary Mathematics, vol. 668, pp. 164–168 (2016)
Léandre, R.: The stochastic flow theorem for an operator of order four. In: Nielsen, F., et al. (eds.) Geometry of Science Information 2013, pp. 497–502. L.N.C.S. 8085, Springer, Berlin (2013)
Léandre, R.: Varadhan estimates for an operator of order four on a Lie group. In: Viedma, E.D., et al. (eds.) Control, Decision and Information Technologies, IEEE, pp. 15. Los-Alamitos, IEEE-Xplore (2016)
Léandre, R.: Wentzel–Freidlin estimates for an operator of order four. In: Akhgar, B., et al. (eds.) 2014 International Conference on Computer Science and Computer Intel, pp. 360–364. IEEE Computer Society. IEEE-Xplore, Los-Alamitos (2014)
Léandre, R.: Malliavin calculus of Bismut type in semi-group theory. Far East J. Math. Sci. 30,n, 1–26 (2008)
Léandre, R.: Stochastic analysis for a non-Markovian generator: an introduction. Russ. J. Math. Phys. 22, 39–52 (2015)
Malliavin, P.: Stochastic calculus of variations and hypoelliptic operators. In: Itô, K. (ed.) Proceedings of International Symposium Stochastic Differential Equations, Kyoto, pp. 195–263. Kinokuyina, Tokyo (1978)
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Léandre, R. Malliavin calculus of Bismut type for an operator of order four on a Lie group. J. Pseudo-Differ. Oper. Appl. 9, 419–430 (2018). https://doi.org/10.1007/s11868-017-0190-3
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DOI: https://doi.org/10.1007/s11868-017-0190-3