Hostname: page-component-7b7567c8c-mnklg Total loading time: 0 Render date: 2026-11-12T19:07:53.884Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic Taylor expansions and heat kernel asymptotics∗

Published online by Cambridge University Press:  31 October 2012

Fabrice Baudoin
Fabrice Baudoin*
Affiliation:
Department of Mathematics Purdue University, 504 Northwestern, Avenu West Lafayette, Indiana, USA. fbaudoin@math.purdue.edu

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

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.

Keywords

Stochastic Taylor expansionsIndex theorems

Information

Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 16 , 2012, pp. 453 - 478
Copyright
© EDP Sciences, SMAI, 2012

References

Références

Atiyah, M.F. and Bott, R., A Lefschetz fixed point formula for elliptic complexes. I. Ann. Math. 86 (1967) 374–407. Google Scholar
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) 237–284. Google Scholar
Azencott, R., Densité des diffusions en temps petit : développements asymptotiques (part I), Sem. Prob. 18 (1984) 402–498. Google Scholar
F. Baudoin, An Introduction to the Geometry of Stochastic Flows. Imperial College Press (2004).
Baudoin, F., Brownian Chen series and Atiyah–Singer theorem. J. Funct. Anal. 254 (2008) 301–317. Google Scholar
Baudoin, F. and Coutin, L., Operators associated with a stochastic differential equation driven by fractional Brownian motions. Stoc. Proc. Appl. 117 (2007) 550–574. Google Scholar
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) 125–153. Google Scholar
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) 307–331. Google Scholar
Ben Arous, G., Développement asymptotique du noyau de la chaleur hypoelliptique sur la diagonale. Ann. Inst. Fourier 39 (1989) 73–99. Google Scholar
Ben Arous, G., Flots et séries de Taylor stochastiques. J. Probab. Theory Relat. Fields 81 (1989) 29–77. Google Scholar
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) 377–402. Google Scholar
N. Berline, E. Getzler and M. Vergne, Heat kernels and Dirac operators, 2nd edition. Grundlehren Text Editions, Springer (2003).
Bismut, J.M., The Atiyah–Singer Theorems : A Probabilistic Approach. J. Func. Anal., Part I, II 57 (1984) 329–348. Google Scholar
N. Bourbaki, Groupes et Algèbres de Lie, Chap. 1–3. Hermann (1972).
Castell, F., Asymptotic expansion of stochastic flows. Probab. Theory Relat. Fields 96 (1993) 225–239. Google Scholar
K.T. Chen, Integration of paths, Geometric invariants and a generalized Baker-Hausdorff formula. Ann. Math. 65 (1957).
Chern, S.S., A simple intrinsic proof of the Gauss-Bonnet theorem for closed Riemannian manifolds. Ann. Math. 45 (1944) 747–752. Google Scholar
Dynkin, E.B., Calculation of the coefficients in the Campbell-Hausdorff formula. Dodakly Akad. Nauk SSSR 57 (1947) 323–326, in Russian, English translation (1997). Google Scholar
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).
A. Friedman, Partial differential equations of parabolic type. Prentice-Hall, Inc., Englewood Cliffs, NJ (1964) xiv+347.
Friz, P. and Victoir, N., Euler estimates for rough differential equations. J. Differ. Equ. 244 (2008) 388–412. Google Scholar
P. Friz and N. Victoir, Multidimensional stochastic processes as rough paths. Theory and Applications, Cambridge Studies in Adv. Math. (2009).
Gaveau, B., Principe de moindre action, propagation de la chaleur et estimées sous-elliptiques sur certains groupes nilpotents. Acta Math. 139 (1977) 95–153. Google Scholar
Getzler, E., A short proof of the Atiyah–Singer index theorem. Topology 25 (1986) 111–117. Google Scholar
Gilkey, P.B., Curvature and the eigenvalues of the Laplacian for elliptic complexes. Adv. Math. 10 (1973) 344–382. Google Scholar
E.P. Hsu, Stochastic Analysis on manifolds. AMS, Providence USA. Grad. Texts Math. 38 (2002).
Y. Inahama, A stochastic Taylor-like expansion in the rough path theory. Preprint from Tokyo Institute of Technology (2007)
P.E. Kloeden and E. Platen, Numerical solution of stochastic differential equations. Appl. Math. 23 (1992).
Kunita, H., Asymptotic self-similarity and short time asymptotics of stochastic flows. J. Math. Sci. Univ. Tokyo 4 (1997) 595–619. Google Scholar
Léandre, R., Sur le théorème d’Atiyah–Singer. Probab. Theory Relat. Fields 80 (1988) 119–137. Google Scholar
Léandre, R., Développement asymptotique de la densité d’une diffusion dégénérée. Forum Math. 4 (1992) 45–75. Google Scholar
Lyons, T., Differential equations driven by rough signals. Revista Mathemàtica Iberio Americana 14 (1998) 215–310. Google Scholar
Lyons, T. and Victoir, N., Cubature on Wiener space. Proc. R. Soc. Lond. A 460 (2004) 169–198. Google Scholar
McKean, H. and Singer, I.M., Curvature and the eigenvalues of the Laplacian. J. Differ. Geom. 1 (1967) 43–69. Google Scholar
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.
P. Malliavin, Stochastic Analysis. Grundlehren der Mathematischen Wissenschaften 313 (1997).
Patodi, V.K., An analytic proof of the Riemann-Roch-Hirzebruch theorem. J. Differ. Geom. 5 (1971) 251–283. Google Scholar
C. Reutenauer, Free Lie algebras, London Mathematical Society Monographs. New series 7 (1993).
S. Rosenberg, The Laplacian on a Riemannian manifold. London Mathematical Society Student Texts 31 (1997).
Rotschild, L.P. and Stein, E.M., Hypoelliptic differential operators and Nilpotent groups. Acta Math. 137 (1976) 247–320. Google Scholar
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.
Strichartz, R.S., The Campbell-Baker-Hausdorff-Dynkin formula and solutions of differential equations. J. Func. Anal. 72. (1987) 320–345. Google Scholar
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) 169–203. Google Scholar
M.E. Taylor, Partial Differential Equations, Basic Theory, 2nd edition. Appl. Math. 23 (1999)
M.E. Taylor, Partial Differential Equations, Qualitative Studies of Linear Equations. Appl. Math. Sci. 116 (1996).