Skip to main content Accessibility help
×
Home

Stochastic Taylor expansions and heat kernel asymptotics

  • Fabrice Baudoin (a1)

Abstract

These notes focus on the applications of the stochastic Taylor expansion of solutions of stochastic differential equations to the study of heat kernels in small times. As an illustration of these methods we provide a new heat kernel proof of the Chern–Gauss–Bonnet theorem.

Copyright

References

Hide All
[1] Atiyah, M.F. and Bott, R., A Lefschetz fixed point formula for elliptic complexes. I. Ann. Math. 86 (1967) 374407.
[2] Azencott, R., Formule de Taylor stochastique et développements asymptotiques d’intégrales de Feynman, in Séminaire de probabilités XVI, edited by J. Azema, M. Yor. Lect. Notes. Math. 921 (1982) 237284.
[3] Azencott, R., Densité des diffusions en temps petit : développements asymptotiques (part I), Sem. Prob. 18 (1984) 402498.
[4] F. Baudoin, An Introduction to the Geometry of Stochastic Flows. Imperial College Press (2004).
[5] Baudoin, F., Brownian Chen series and Atiyah–Singer theorem. J. Funct. Anal. 254 (2008) 301317.
[6] Baudoin, F. and Coutin, L., Operators associated with a stochastic differential equation driven by fractional Brownian motions. Stoc. Proc. Appl. 117 (2007) 550574.
[7] Ben Arous, G., Méthodes de Laplace et de la phase stationnaire sur l’espace de Wiener (French) [The Laplace and stationary phase methods on Wiener space]. Stochastics 25 (1988) 125153.
[8] Ben Arous, G., Développement asymptotique du noyau de la chaleur hypoelliptique hors du cut-locus (French) [Asymptotic expansion of the hypoelliptic heat kernel outside of the cut-locus]. Ann. Sci. Cole Norm. Sup. 21 (1988) 307331.
[9] Ben Arous, G., Développement asymptotique du noyau de la chaleur hypoelliptique sur la diagonale. Ann. Inst. Fourier 39 (1989) 7399.
[10] Ben Arous, G., Flots et séries de Taylor stochastiques. J. Probab. Theory Relat. Fields 81 (1989) 2977.
[11] Ben Arous, G. and Léandre, R., Décroissance exponentielle du noyau de la chaleur sur la diagonale. II (French) [Exponential decay of the heat kernel on the diagonal II] Probab. Theory Relat. Fields 90 (1991) 377402.
[12] N. Berline, E. Getzler and M. Vergne, Heat kernels and Dirac operators, 2nd edition. Grundlehren Text Editions, Springer (2003).
[13] Bismut, J.M., The Atiyah–Singer Theorems : A Probabilistic Approach. J. Func. Anal., Part I, II 57 (1984) 329348.
[14] N. Bourbaki, Groupes et Algèbres de Lie, Chap. 1–3. Hermann (1972).
[15] Castell, F., Asymptotic expansion of stochastic flows. Probab. Theory Relat. Fields 96 (1993) 225239.
[16] K.T. Chen, Integration of paths, Geometric invariants and a generalized Baker-Hausdorff formula. Ann. Math. 65 (1957).
[17] Chern, S.S., A simple intrinsic proof of the Gauss-Bonnet theorem for closed Riemannian manifolds. Ann. Math. 45 (1944) 747752.
[18] Dynkin, E.B., Calculation of the coefficients in the Campbell-Hausdorff formula. Dodakly Akad. Nauk SSSR 57 (1947) 323326, in Russian, English translation (1997).
[19] M. Fliess and D. Normand-Cyrot, Algèbres de Lie nilpotentes, formule de Baker-Campbell-Hausdorff et intégrales itérées de K.T. Chen, in Séminaire de Probabilités. Lect. Notes Math. 920 (1982).
[20] A. Friedman, Partial differential equations of parabolic type. Prentice-Hall, Inc., Englewood Cliffs, NJ (1964) xiv+347.
[21] Friz, P. and Victoir, N., Euler estimates for rough differential equations. J. Differ. Equ. 244 (2008) 388412.
[22] P. Friz and N. Victoir, Multidimensional stochastic processes as rough paths. Theory and Applications, Cambridge Studies in Adv. Math. (2009).
[23] Gaveau, B., Principe de moindre action, propagation de la chaleur et estimées sous-elliptiques sur certains groupes nilpotents. Acta Math. 139 (1977) 95153.
[24] Getzler, E., A short proof of the Atiyah–Singer index theorem. Topology 25 (1986) 111117.
[25] Gilkey, P.B., Curvature and the eigenvalues of the Laplacian for elliptic complexes. Adv. Math. 10 (1973) 344382.
[26] E.P. Hsu, Stochastic Analysis on manifolds. AMS, Providence USA. Grad. Texts Math. 38 (2002).
[27] Y. Inahama, A stochastic Taylor-like expansion in the rough path theory. Preprint from Tokyo Institute of Technology (2007)
[28] P.E. Kloeden and E. Platen, Numerical solution of stochastic differential equations. Appl. Math. 23 (1992).
[29] Kunita, H., Asymptotic self-similarity and short time asymptotics of stochastic flows. J. Math. Sci. Univ. Tokyo 4 (1997) 595619.
[30] Léandre, R., Sur le théorème d’Atiyah–Singer. Probab. Theory Relat. Fields 80 (1988) 119137.
[31] Léandre, R., Développement asymptotique de la densité d’une diffusion dégénérée. Forum Math. 4 (1992) 4575.
[32] Lyons, T., Differential equations driven by rough signals. Revista Mathemàtica Iberio Americana 14 (1998) 215310.
[33] Lyons, T. and Victoir, N., Cubature on Wiener space. Proc. R. Soc. Lond. A 460 (2004) 169198.
[34] McKean, H. and Singer, I.M., Curvature and the eigenvalues of the Laplacian. J. Differ. Geom. 1 (1967) 4369.
[35] P. Malliavin, Stochastic calculus of variations and hypoelliptic operators, in Proc. of Inter. Symp. Stoch. Differ. Equ., Kyoto 1976, edited by Wiley (1978) 195–263.
[36] P. Malliavin, Stochastic Analysis. Grundlehren der Mathematischen Wissenschaften 313 (1997).
[37] Patodi, V.K., An analytic proof of the Riemann-Roch-Hirzebruch theorem. J. Differ. Geom. 5 (1971) 251283.
[38] C. Reutenauer, Free Lie algebras, London Mathematical Society Monographs. New series 7 (1993).
[39] S. Rosenberg, The Laplacian on a Riemannian manifold. London Mathematical Society Student Texts 31 (1997).
[40] Rotschild, L.P. and Stein, E.M., Hypoelliptic differential operators and Nilpotent groups. Acta Math. 137 (1976) 247320.
[41] D. Stroock and S.R.S. Varadhan, Multidimensional diffusion processes. Springer-Verlag, Berlin, New York. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 233 (1979) xii+338.
[42] Strichartz, R.S., The Campbell-Baker-Hausdorff-Dynkin formula and solutions of differential equations. J. Func. Anal. 72. (1987) 320345.
[43] Takanobu, S., Diagonal short time asymptotics of heat kernels for certain degenerate second order differential operators of Hörmander type. Publ. Res. Inst. Math. Sci. 24 (1988) 169203.
[44] M.E. Taylor, Partial Differential Equations, Basic Theory, 2nd edition. Appl. Math. 23 (1999)
[45] M.E. Taylor, Partial Differential Equations, Qualitative Studies of Linear Equations. Appl. Math. Sci. 116 (1996).

Keywords

Related content

Powered by UNSILO

Stochastic Taylor expansions and heat kernel asymptotics

  • Fabrice Baudoin (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.