Summary
We give examples based upon large deviation's theory where the heat kernel of a degenerate diffusion has an exponential decay over the diagonal. Using Malliavin calculus, we give conditions for a more generalized heat kernel to have an exponential decay over the diagonal. We give lower bound in some particular case by using the Bismut's condition.
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[Az.1] Azencott, R.: Formule de Taylor stochastique et développement asymptotique d'intégrales de Feynman. Séminaire de Probabilités XVI. (Lect. Notes Math., vol. 921) Berlin Heidelberg New York: Springer 1980/81
[Az.2] Azencott, R.: Grandes déviations et applications. Cours de probabilité de Saint-Flour. (Lect. Notes Math., vol. 774) Berlin Heidelberg New York: Springer 1978
[BA.4] Ben Arous, G.: Noyau de la chaleur hypoelliptique et géométrique sous-riemannienne. Actes du Colloque Franco-Japonais, Paris. (Lect. Notes Math., vol. 1322) Berlin Heidelberg New York: Springer 1987
[BA.2] Ben Arous, G.: Methodes de Laplace et de la phase stationnaire sur l'espace de Wiener. Stochastics25, 125–153 (1988)
[BA.3] Ben Arous, G.: Developpement asymptotique du noyau de la chaleur hors du cutlocus. Ann. Sci. Ec. Norm. Super., IV, Ser.21, 307–331 (1988)
[BA.1] Ben Arous, G.: Flots et séries de Taylor stochastiques. Probab. Th. Rel. Fields81, 29–77 (1989)
[BA-L.1] Ben Arous, G., Léandre, R.: Influence du drift sur le comportement au point de départ d'un noyau de la chaleur hypoelliptique (II). A paraître à Probab. Th. Rel. Fields
[BA-L.2] Ben Arous, G., Léandre, R.: Formule de Campbell-Hausdorff-Dynkin stochastique. (preprint)
[B.1] Bismut, J.-M.: in Large deviations and the Malliavin calculus. In: Progress in Maths, vol. 45. Boston: Birkhaüser 1984
[B.2] Bismut, J-M.: Martingales, the Malliavin calculus and hypoellipticity under general Hörmander's conditions. Z. Wahrscheinlichkeitstheor. Vew. Geo.56, 469–505 (1981)
[F-V] Freidlin, M.I. Ventcel, A.D.: Random perturbation of dynamical system. Grundlehren der Mathématischen Wissenschaften, vol. 260 Berlin Heidelberg New York: Springer 1984
[J-S.1] Jerison, D., Sanchez-Calle, A.: Subelliptic second order differential operator in complex analysis III. In: Berenstein, E. (Ed.) (Lect. Notes Math., vol. 1227, pp. 46–78) Berlin Heidelberg New York: Springer 1987
[J-S.2] Jerison, D., Sanchez-Calle, A.: Estimates for the heat kernel for a sum of squares of vector fields. Indiana Univ. Math. J.35, 835–854 (1986)
[K-S.1] Kusuoka, S., Stroock, D.W.: Long time estimates for the heat kernel associated with uniformly subelliptic symmetric second order operator. Ann. Math.127, 165–189 (1989)
[K-S.2] Kusuoka, S., Stroock, D.W.: Applications of the Malliavin calculus II
[K-S.3] Kusuoka, S., Stroock, D.W.: Applications of the Malliavin calculus III
[L.1] Léandre, R.: Applications quantitatives et géométriques du calcul de Malliavin. Version française: Actes du Colloque Franco-Japonais. (Lect. Notes Math., vol. 1322) Berlin Heidelberg New York: Springer Version anglaise: Proceedings du Colloque Geometry of random motion. A.M.S. Contemp. Math.73, 173–196 (1989)
[L.2] Léandre, R.: Minoration en temps petit de la densité d'une diffusion dégénérée. J. Funct. Anal.74, 399–415 (1987)
[L.3] Léandre, R.: Développementasymptotique de la densité d'une diffusion dégénérée. A paraitre dans Forum Mathematicum (1991)
[M] Meyer, P.-A.: Le calcul de Malliavin et un peu de pédagogie. RCP no 25, vol. 34, Université de Strasbourg, 1984
[N-S-W] Nagel, A., Stein, E.M., Wainger, S.: Balls and metrics defined by vector fields, I. Basic Properties. Acta Math.155, 103–147 (1985)
[Str] Strichartz, R.: The Campbell-Baker-Hausdorff-Dynkin formula and solutions of differential equations. J. Funct. Anal.72, 320–346 (1987)
[T] Takanobu, S.: Diagonal short time asymptotics of heat kernels for certain degenerate operators of Hörmander type. Publ. Res. Inst. Math. Sci.24, 169–203 (1988)
[V] Varopoulos, N.: (preprint)
[W] Watanabe, S.: Analysis of Wiener functional and its applications to heat kernels. Ann. Probab.15, 1–39 (1987)
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Ben Arous, G., Léandre, R. Decroissance exponentielle du noyau de la chaleur sur la diagonale (I). Probab. Th. Rel. Fields 90, 175–202 (1991). https://doi.org/10.1007/BF01192161
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DOI: https://doi.org/10.1007/BF01192161