Stochastic Taylor expansions and heat kernel asymptotics

 Access to full text

1. [1] M.F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes. I. Ann. Math.86 (1967) 374–407. Zbl0161.43201MR212836
2. [2] R. Azencott, 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. Zbl0484.60064MR658728
3. [3] R. Azencott, Densité des diffusions en temps petit : développements asymptotiques (part I), Sem. Prob.18 (1984) 402–498. Zbl0546.60079MR770974
4. [4] F. Baudoin, An Introduction to the Geometry of Stochastic Flows. Imperial College Press (2004). Zbl1085.60002MR2154760
5. [5] F. Baudoin, Brownian Chen series and Atiyah–Singer theorem. J. Funct. Anal.254 (2008) 301–317. Zbl1147.58017MR2376573
6. [6] F. Baudoin and L. Coutin, Operators associated with a stochastic differential equation driven by fractional Brownian motions. Stoc. Proc. Appl.117 (2007) 550–574. Zbl1119.60043MR2320949
7. [7] G. Ben Arous, Méthodes de Laplace et de la phase stationnaire sur l’espace de Wiener (French) [The Laplace and stationary phase methods on Wiener space]. Stochastics25 (1988) 125–153. Zbl0666.60026MR999365
8. [8] G. Ben Arous, 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. Zbl0699.35047MR974408
9. [9] G. Ben Arous, Développement asymptotique du noyau de la chaleur hypoelliptique sur la diagonale. Ann. Inst. Fourier39 (1989) 73–99. Zbl0659.35024MR1011978
10. [10] G. Ben Arous, Flots et séries de Taylor stochastiques. J. Probab. Theory Relat. Fields81 (1989) 29–77. Zbl0639.60062MR981567
11. [11] G. Ben Arous and R. Léandre, 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. Zbl0734.60027MR1133372
12. [12] N. Berline, E. Getzler and M. Vergne, Heat kernels and Dirac operators, 2nd edition. Grundlehren Text Editions, Springer (2003). Zbl1037.58015MR2273508
13. [13] J.M. Bismut, The Atiyah–Singer Theorems : A Probabilistic Approach. J. Func. Anal., Part I, II 57 (1984) 329–348. Zbl0556.58027MR756173
14. [14] N. Bourbaki, Groupes et Algèbres de Lie, Chap. 1–3. Hermann (1972). Zbl0483.22001
15. [15] F. Castell, Asymptotic expansion of stochastic flows. Probab. Theory Relat. Fields96 (1993) 225–239. Zbl0794.60054MR1227033
16. [16] K.T. Chen, Integration of paths, Geometric invariants and a generalized Baker-Hausdorff formula. Ann. Math. 65 (1957). Zbl0077.25301MR85251
17. [17] S.S. Chern, A simple intrinsic proof of the Gauss-Bonnet theorem for closed Riemannian manifolds. Ann. Math.45 (1944) 747–752. Zbl0060.38103MR11027
18. [18] E.B. Dynkin, Calculation of the coefficients in the Campbell-Hausdorff formula. Dodakly Akad. Nauk SSSR 57 (1947) 323–326, in Russian, English translation (1997). Zbl0029.24507MR21940
19. [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). Zbl0495.60064
20. [20] A. Friedman, Partial differential equations of parabolic type. Prentice-Hall, Inc., Englewood Cliffs, NJ (1964) xiv+347. Zbl0144.34903MR181836
21. [21] P. Friz and N. Victoir, Euler estimates for rough differential equations. J. Differ. Equ.244 (2008) 388–412. Zbl1140.60037MR2376201
22. [22] P. Friz and N. Victoir, Multidimensional stochastic processes as rough paths. Theory and Applications, Cambridge Studies in Adv. Math. (2009). Zbl1193.60053MR2604669
23. [23] B. Gaveau, Principe de moindre action, propagation de la chaleur et estimées sous-elliptiques sur certains groupes nilpotents. Acta Math.139 (1977) 95–153. Zbl0366.22010MR461589
24. [24] E. Getzler, A short proof of the Atiyah–Singer index theorem. Topology25 (1986) 111–117. Zbl0607.58040MR836727
25. [25] P.B. Gilkey, Curvature and the eigenvalues of the Laplacian for elliptic complexes. Adv. Math.10 (1973) 344–382. Zbl0259.58010MR324731
26. [26] E.P. Hsu, Stochastic Analysis on manifolds. AMS, Providence USA. Grad. Texts Math. 38 (2002). Zbl0994.58019MR1882015
27. [27] Y. Inahama, A stochastic Taylor-like expansion in the rough path theory. Preprint from Tokyo Institute of Technology (2007) Zbl1203.60073MR2679952
28. [28] P.E. Kloeden and E. Platen, Numerical solution of stochastic differential equations. Appl. Math. 23 (1992). Zbl0752.60043MR1214374
29. [29] H. Kunita, Asymptotic self-similarity and short time asymptotics of stochastic flows. J. Math. Sci. Univ. Tokyo4 (1997) 595–619. Zbl0890.60007MR1484603
30. [30] R. Léandre, Sur le théorème d’Atiyah–Singer. Probab. Theory Relat. Fields80 (1988) 119–137. Zbl0639.58024
31. [31] R. Léandre, Développement asymptotique de la densité d’une diffusion dégénérée. Forum Math.4 (1992) 45–75. Zbl0749.60054
32. [32] T. Lyons, Differential equations driven by rough signals. Revista Mathemàtica Iberio Americana14 (1998) 215–310. Zbl0923.34056MR1654527
33. [33] T. Lyons and N. Victoir, Cubature on Wiener space. Proc. R. Soc. Lond. A460 (2004) 169–198. Zbl1055.60049MR2052260
34. [34] H. McKean and I.M. Singer, Curvature and the eigenvalues of the Laplacian. J. Differ. Geom.1 (1967) 43–69. Zbl0198.44301MR217739
35. [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. Zbl0411.60060MR536013
36. [36] P. Malliavin, Stochastic Analysis. Grundlehren der Mathematischen Wissenschaften 313 (1997). Zbl0878.60001MR1450093
37. [37] V.K. Patodi, An analytic proof of the Riemann-Roch-Hirzebruch theorem. J. Differ. Geom.5 (1971) 251–283. Zbl0219.53054MR290318
38. [38] C. Reutenauer, Free Lie algebras, London Mathematical Society Monographs. New series 7 (1993). Zbl0798.17001MR1231799
39. [39] S. Rosenberg, The Laplacian on a Riemannian manifold. London Mathematical Society Student Texts 31 (1997). Zbl0868.58074MR1462892
40. [40] L.P. Rotschild and E.M. Stein, Hypoelliptic differential operators and Nilpotent groups. Acta Math.137 (1976) 247–320. Zbl0346.35030MR436223
41. [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. Zbl0426.60069MR532498
42. [42] R.S. Strichartz, The Campbell-Baker-Hausdorff-Dynkin formula and solutions of differential equations. J. Func. Anal. 72. (1987) 320–345. Zbl0623.34058MR886816
43. [43] S. Takanobu, 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. Zbl0677.35019MR944857
44. [44] M.E. Taylor, Partial Differential Equations, Basic Theory, 2nd edition. Appl. Math. 23 (1999) Zbl0869.35001MR1395147
45. [45] M.E. Taylor, Partial Differential Equations, Qualitative Studies of Linear Equations. Appl. Math. Sci. 116 (1996). Zbl0869.35003MR1395149