Elworthy, K. D. and Li, X-M. (Xue-Mei), 1964-. (2008) An L-2 theory for differential forms on path spaces I. Journal of Functional Analysis, Vol.254 (No.1). pp. 196-245. ISSN 0022-1236

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Abstract

An L-2 theory of differential forms is proposed for the Banach manifold of continuous paths on a Riemannian manifold M furnished with its Brownian motion measure. Differentiation must be restricted to certain Hilbert space directions, the H-tangent vectors. To obtain a closed exterior differential operator the relevant spaces of differential forms, the H-forms, are perturbed by the curvature of M. A Hodge decomposition is given for L-2 H-one-forms, and the structure of H-two-forms is described. The dual operator d* is analysed in terms of a natural connection on the H-tangent spaces. Malliavin calculus is a basic tool. (c) 2007 Elsevier Inc. All rights reserved.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Differential forms, Malliavin calculus, Banach manifolds, Curvature, Infinite-dimensional manifolds, Hodge theory, Homology theory
Journal or Publication Title:	Journal of Functional Analysis
Publisher:	Academic Press
ISSN:	0022-1236
Date:	1 January 2008
Volume:	Vol.254
Number:	No.1
Number of Pages:	50
Page Range:	pp. 196-245
Identification Number:	10.1016/j.jfa.2007.09.016
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	Engineering and Physical Sciences Research Council (EPSRC), European Union (EU), Royal Society (Great Britain), National Science Foundation (U.S.) (NSF), Alexander von Humboldt-Stiftung
Grant number:	GR/NOO 845 (EPSRC), ERB-FMRX-CT96- 0075 (EU), DMS 0072387 (NSF)
References:	[1] Z.M. Ahmed, D.W. Stroock, A Hodge theory for some non-compact manifolds, J. Differential Geom. 54 (1) (2000) 177–225. [2] S. Aida, On the irreducibility of certain Dirichlet forms on loop spaces over compact homogeneous spaces, in: K.D. Elworthy, S. Kusuoka, I. Shigekawa (Eds.), New Trends in Stochastic Analysis, Proc. Taniguchi Symposium, Charingworth, September 1995, World Sci. Press, 1997. [3] H. Airault, Subordination de processus dans le fibre tangent et formes harmoniques, C. R. Acad. Sci. Paris Sér. A 282 (1976) 1311–1314. [4] H. Airault, J. van Biesen, Géométrie riemannienne en codimension finie sur l’espace de Wiener, C. R. Acad. Sci. Paris Sér. I Math. 311 (2) (1990) 125–130. [5] A. Arai, I. Mitoma, de Rham–Hodge–Kodaira decomposition in ∞-dimensions, Math. Ann. 291 (1) (1991) 51–73. [6] M.F. Atiyah, Elliptic operators, discrete groups, and von Neumann algebras, Astérisque 32–33 (1976) 43–72. [7] C.J. Atkin, Extension of smooth functions in infinite dimensions. I. Unions of convex sets, Studia Math. 146 (3) (2001) 201–226. [8] C.J. Atkin, Extension of smooth functions in infinite dimensions. II. Manifolds, Studia Math. 150 (3) (2002) 215– 241. [9] D. Bell, Divergence theorems in path space, J. Funct. Anal. 218 (1) (2005) 130–149. [10] J.-M. Bismut, Martingales, the Malliavin calculus and Hörmander’s theorem, in: Stochastic Integrals (Proc. Sympos.), Durham, 1980, in: Lecture Notes in Math., vol. 851, Springer-Verlag, 1981, pp. 85–109. [11] R. Bonic, J. Frampton, A. Tromba, λ-Manifolds, J. Funct. Anal. 3 (1969) 310–320. [12] Ed. Bueler, I. Prokhorenkov, Hodge theory and cohomology with compact supports, Soochow J.Math. 28 (1) (2002) 33–55. [13] R. Carmona, S. Chevet, Tensor Gaussian measures on Lp(E), J. Funct. Anal. 33 (1979) 297–310. [14] J. Cigler, V. Losert, P. Michor, Banach Modules and Functors on Categories of Banach Spaces, Lect. Notes Pure Appl. Math., vol. 46, Marcel Dekker, Inc. XV, New York, Basel, 1979. [15] A. Cruzeiro, S. Fang, Une inégalité L2 pour des intégrales stochastiques anticipatives sur une variété riemannienne, C. R. Acad. Sci. Paris Sér. I 321 (1995) 1245–1250. [16] A. Cruzeiro, S. Fang, Weak Levi-Civita connection for the damped metric on the Riemannian path space and vanishing of Ricci tensor in adapted differential geometry, J. Funct. Anal. 185 (2) (2001) 681–698. [17] A. Cruzeiro, S. Fang, P. Malliavin, A probabilistic Weitzenböck formula on Riemannian path space, J. Anal. Math. 80 (2000) 87–100. [18] A. Cruzeiro, P. Malliavin, Renormalized differential geometry on path space: Structural equation, curvature, J. Funct. Anal. 139 (1) (1996) 119–181. [19] A. Cruzeiro, P.Malliavin, A class of anticipative tangent processes on the Wiener space, C. R. Acad. Sci. Paris Sér. I Math. 333 (4) (2001) 353–358. [20] B.K. Driver, A Cameron–Martin type quasi-invariance theorem for Brownian motion on a compact Riemannian manifold, J. Funct. Anal. 100 (1992) 272–377. [21] B.K. Driver, The Lie bracket of adapted vector fields onWiener spaces, Appl. Math. Optim. 39 (2) (1999) 179–210. [22] B.K. Driver, A. Thalmaier, Heat equation derivative formulas for vector bundles, J. Funct. Anal. 183 (1) (2001) 42–108. [23] A. Eberle, Uniqueness and Non-Uniqueness of Semigroups Generated by Singular Diffusion Operators, Lecture Notes in Math., vol. 1718, Springer-Verlag, 1999. [24] J. Eells Jr., A setting for global analysis, Bull. Amer. Math. Soc. 72 (1966) 751–807. [25] H.I. Eliasson, Geometry of manifolds of maps, J. Differential Geom. 1 (1967) 169–194. [26] K.D. Elworthy, Geometric aspects of diffusions on manifolds, in: P.L. Hennequin (Ed.), Ecole d’Eté de Probabilités de Saint-Flour XV–XVII, 1985–1987, in: Lecture Notes in Math., vol. 1362, Springer-Verlag, 1988, pp. 276–425. [27] K.D. Elworthy, Stochastic flows on Riemannian manifolds, in:M.A. Pinsky, V.Wihstutz (Eds.), Diffusion Processes and Related Problems in Analysis, vol. II, in: Progr. Probab., Birkhäuser, Boston, 1992, pp. 37–72. [28] K.D. Elworthy, Y. LeJan, Xue-Mei Li, Integration by parts formulae for degenerate diffusion measures on path spaces and diffeomorphism groups, C. R. Acad. Sci. Paris Sér. I 323 (1996) 921–926. [29] K.D. Elworthy, Y. LeJan, Xue-Mei Li, Concerning the geometry of stochastic differential equations and stochastic flows, in: K.D. Elworthy, S. Kusuoka, I. Shigekawa (Eds.), New Trends in Stochastic Analysis, Proc. Taniguchi Symposium, Charingworth, September 1995, World Scientific Press, 1997. [30] K.D. Elworthy, Y. LeJan, Xue-Mei Li, On the Geometry of Diffusion Operators and Stochastic Flows, Lecture Notes in Math., vol. 1720, Springer-Verlag, 1999. [31] K.D. Elworthy, Xue-Mei Li, Bismut type formulae for differential forms, C. R. Acad. Sci. Paris Sér. I Math. 327 (1) (1998) 87–92. [32] K.D. Elworthy, Xue-Mei Li, Special Itô maps and an L2 Hodge theory for one forms on path spaces, Canadian Math. Soc. Conf. Proc. 28 (2000) 145–162. [33] K.D. Elworthy, Xue-Mei Li, Some families of q-vector fields on path spaces, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (Suppl.) (2003) 1–27. [34] K.D. Elworthy, Xue-Mei Li, Geometric stochastic analysis on path spaces, in: Proceedings of the International Congress of Mathematicians, Madrid, 2006, pp. 575–594. [35] K.D. Elworthy, Xue-Mei Li, Itô maps and analysis on path spaces, Math. Z. 257 (3) (2007). [36] K.D. Elworthy, Xue-Mei Li, S. Rosenberg, Bounded and L2 harmonic forms on universal covers, Geom. Funct. Anal. 8 (1998) 283–303. [37] K.D. Elworthy, Zhi-Ming Ma, Vector fields on mapping spaces and related Dirichlet forms and diffusions, Osaka J. Math. 34 (3) (1997) 629–651. [38] K.D. Elworthy, M. Yor, Conditional expectations for derivatives of certain stochastic flows, in: J. Azéma, P.A. Meyer, M. Yor (Eds.), Séminaire de Probabilités XXVII, in: Lecture Notes in Math., vol. 1557, Springer-Verlag, 1993, pp. 159–172. [39] S. Fang, Inégalité du type de Poincaré sur l’espace des chemins riemanniens, C. R. Acad. Sci. Paris Sér. I Math. 318 (3) (1994) 257–260. [40] S. Fang, Stochastic anticipative integrals on a Riemannian manifold, J. Funct. Anal. 131 (1) (1995) 228–253. [41] S. Fang, Stochastic anticipative calculus on the path space over a compact Riemannian manifold, J. Math. Pures Appl. (9) 77 (3) (1998) 249–282. [42] S. Fang, J. Franchi, De Rham–Hodge–Kodaira operator on loop groups, J. Funct. Anal. 148 (1997) 391–407. [43] Fu-Zhou Gong, Feng-Yu Wang, On Gromov’s theorem and L2-Hodge decomposition, Int. J. Math. Math. Sci. 2004 (1–4) (2004) 25–44. [44] L. Gross, Potential theory on Hilbert space, J. Funct. Anal. 1 (1967) 123–181. [45] Y. Hu, A.S. Üstünel, M. Zakai, Tangent processes on Wiener space, J. Funct. Anal. 192 (1) (2002) 234–270. [46] N. Ikeda, S.Watanabe, Stochastic Differential Equations and Diffusion Processes, second ed., North-Holland, 1989. [47] J.D.S. Jones, R. Léandre, Lp-Chen forms on loop spaces, in: Stochastic Analysis, Durham, 1990, in: London Math. Soc. Lecture Note Ser., vol. 167, Cambridge University Press, Cambridge, 1991, pp. 103–162. [48] T. Kazumi, I. Shigekawa, Differential calculus on a submanifold of an abstract Wiener space. II. Weitzenböck formula, in: Dirichlet Forms and Stochastic Processes, Beijing, 1993, de Gruyter, Berlin, 1995, pp. 235–251. [49] Y. Kifer, A note on integrability of Cr -norms of stochastic flows and applications, in: Stochastic Mechanics and Stochastic Processes, Swansea, 1986, in: Lecture Notes in Math., vol. 1325, Springer-Verlag, 1988, pp. 125–131. [50] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, vol. I, Interscience Publishers, New York, London, 1963. [51] M. Krée, P. Krée, Continuité de la divergence dans les espaces de Sobolev relatifs l’espace de Wiener, C. R. Acad. Sci. Paris Sér. I Math. 296 (20) (1983) 833–836. [52] S. Kusuoka, de Rham cohomology of Wiener–Riemannian manifolds, in: Proceedings of the International Congress of Mathematicians, vols. I, II, Kyoto, 1990, Math. Soc. Japan, Tokyo, 1991, pp. 1075–1082. [53] S. Kusuoka, Analysis on Wiener spaces. II. Differential forms, J. Funct. Anal. 103 (2) (1992) 229–274. [54] S. Lang, Introduction to Differential Manifolds, Interscience Publishers, 1962. [55] R. Léandre, Cohomologie de Bismut–Nualart–Pardoux et cohomologie de Hochschild entière, in: Séminaire de Probabilités XXX, in: Lecture Notes in Math., vol. 1626, Springer-Verlag, 1996, pp. 68–99. [56] R. Léandre, Brownian cohomology of an homogeneous manifold, in: New Trends in Stochastic Analysis, Charingworth, 1994, World Sci. Publishing, 1997, pp. 305–347. [57] R. Léandre, Singular integral homology of the stochastic loop space, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1 (1) (1998) 17–31. [58] R. Léandre, Stochastic cohomology of the frame bundle of the loop space, J. Nonlinear Math. Phys. 5 (1) (1998) 23–40. [59] R. Léandre, Analysis on loop spaces, and topology, Mat. Zametki 72 (2) (2002) 236–257 (in Russian); translated in: Math. Notes 72 (1–2) (2002) 212–229. [60] László Lempert, Ning Zhang, Dolbeault cohomology of a loop space, Acta Math. 193 (2) (2004) 241–268. [61] P. Malliavin, Stochastic Analysis, Grundlehren Math. Wiss., vol. 313, Springer-Verlag, 1997. [62] I. Mitoma, De Rham–Kodaira decomposition and fundamental spaces of Wiener functionals, in: Gaussian Random Fields, Nagoya, 1990, World Sci. Publishing, 1991, pp. 285–297. [63] J.R. Norris, Twisted sheets, J. Funct. Anal. 132 (2) (1995). [64] D. Nualart, The Malliavin Calculus and Related Topics, Springer-Verlag, 1995. [65] M.A. Piech, A model for an infinite-dimensional Laplace Beltrami operator, Indiana Univ. Math. J. 31 (3) (1982) 327–340. [66] A. Pietsch, Operator Ideals, Mathematische Monographien, vol. 16, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. [67] A.F. Ruston, Fredholm Theory in Banach Spaces, Cambridge Tracts in Math., vol. 86, Cambridge University Press, 1986. [68] L. Schwartz, Un théorème de dualité pour les applications radonifiantes, Acad. Sci. Paris Sér. A 268 (1969) 1410– 1413. [69] I. Shigekawa, De Rham–Hodge–Kodaira’s decomposition on an abstract Wiener space, J. Math. Kyoto Univ. 26 (2) (1986) 191–202. [70] I. Shigekawa, Vanishing theorem of the Hodge–Kodaira operator for differential forms on a convex domain of the Wiener space, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (Suppl.) (2003) 53–63. [71] J. van Biesen, The divergence on submanifolds of the Wiener space, J. Funct. Anal. 113 (2) (1993) 426–461.
URI:	http://wrap.warwick.ac.uk/id/eprint/30670

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