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Exponential decay of the heat kernel over the diagonal. II. (Décroissance exponentielle du noyau de la chaleur sur la diagonale. II.) (French) Zbl 0734.60027

[For part I see the preceding review, Zbl 0734.60026.]
We give some conditions for the heat kernel to have an asymptotic expansion in small time such that all coefficients vanish, although the phenomenon seems difficult to understand by large deviations theory. The fact that the leading term is not zero is strongly related to Bismut’s condition. These examples are related to the Varadhan estimates of the density of a dynamical system submitted to small random perturbations. To understand that type of asymptotic, one must modify the definition of the distance by adding the Bismut condition (unnoticed, but hidden, in classical cases).
Reviewer: G.Ben Arous

MSC:

60F10 Large deviations
60J60 Diffusion processes

Citations:

Zbl 0734.60026
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[1] [Az_1] Azencott, R.: Grandes déviations et applications. Cours de probabilités de Saint-Flour. (Lect. Notes Math., vol. 774). Berlin Heidelberg New York: 1978
[2] Azencott, R.; Baldi, P.; Bellaiche, A.; Bellaiche, C.; Bougerol, P.; Chaleyat-Maurel, M.; Elie, L.; Granara, J., Géodésiques et diffusions en temps petit. Société mathématique de France, Astérisque, 84-85, 3-279 (1981)
[3] Bismut, J. M., Large deviations and the Malliavin-Calculus (1984), Basel Boston Stuttgart: Birkhäuser, Basel Boston Stuttgart · Zbl 0537.35003
[4] Bismut, J. M., Mécanique aléatoire (1981), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York
[5] [B.A_1] Ben-Arous, G.: Méthodes de laplace et de la phase stationnaire sur l’espace de Wiener. (Preprint) · Zbl 0666.60026
[6] Ben-Arous, G.; Métivier, M.; Watanabe, S., Noyau de la chaleur hypoelliptique et géométrie sous-riemannienne, Stochastic analysis, 1-17 (1989), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York
[7] [B.A-L] Ben-Arous, G., Léandre, R.; Décroissance exponentielle du noyau de la chaleur sur la diagonale (I). (A paraître au Z.W.) · Zbl 0734.60026
[8] Fernique, X., Intégrabilité des vecteurs gaussiens, C.R. Acad. Sci., Sér. A, 270, 1698-1699 (1970) · Zbl 0206.19002
[9] Freidlin, M. I.; Ventcel, A. D., Random perturbation of dynamical system (1984), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York
[10] Ikeda, N.; Watanabe, S., Stochastic differential equations and diffusion processes (1981), Amsterdam: North-Holland, Amsterdam
[11] Jerison, D.; Sanchez, A.; Berenstein, E., Subelliptic second order differential operator, complex analysis III., 46-78 (1987), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York
[12] Kree, P.; Korezlioglu, H.; Ustunel, S., La théorie des distributions en dimension quelconque et l’intégration stochastique. Stochastic analysis and related topics, Lect. Notes Math., vol. 1316, 170-234 (1989), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York
[13] Kusuoka, S.; Stroock, D. W.; Itô, K., Applications of the Malliavin Calculus. Part I, Stochastic analysis, 271-306 (1981), Tokyo: Kinokuniya, Tokyo
[14] Kusuoka, S.; Stroock, D. W., Applications of the Malliavin Calculus. Part II, J. Fac. Sci. Univ. Tokyo, Sect. IA, 32, 1-76 (1985) · Zbl 0568.60059
[15] Léandre, R.; Métivier, M.; Watanabe, S., Applications quantitatives et géométrique du calcul de Malliavin, Stochastic analysis, 109-134 (1989), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York
[16] Léandre, R., Intégration dans la fibre associée à une diffusion dégénérée, Probab. Théory Relat. Fields, 76, 341-358 (1987) · Zbl 0611.60051
[17] [L_3] Léandre, R.: Développement asymptotique de la densité d’une diffusion dégénérée. (A paraître dans Forum mahtematicum) · Zbl 0749.60054
[18] Léandre, R., Estimation en temps petit de la densité d’une diffusion hypoelliptique, C.R. Acad. Sci. Paris Sér. I., 17, 801-804 (1985) · Zbl 0585.60075
[19] Léandre, R.; Russo, F., Estimation de Varadhan pour des diffusions à deux paramètres, Probab. Théory Relat. Fields, 84, 429-451 (1990) · Zbl 0665.60057
[20] [S.V] Stroock, D.W., Varadhan S.R.S.: On the support of diffusion processes with applications to the strong maximum principle. Sixth Berkeley Symposium, pp. 333-368
[21] Watanabe, S., Analysis of Wiener functionals (Malliavin Calculus) and its applications to heat kernels, Ann. Probab., 15, 1-39 (1987) · Zbl 0633.60077
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