×

Itô maps and analysis on path spaces. (English) Zbl 1181.60078

Summary: We consider versions of Malliavin calculus on path spaces of compact manifolds with diffusion measures, defining Gross-Sobolev spaces of differentiable functions and proving their intertwining with solution maps, \({\mathcal{I}}\), of certain stochastic differential equations. This is shown to shed light on fundamental uniqueness questions for this calculus including uniqueness of the closed derivative operator \(d\) and Markov uniqueness of the associated Dirichlet form. A continuity result for the divergence operator by Kree and Kree is extended to this situation. The regularity of conditional expectations of smooth functionals of classical Wiener space, given \({\mathcal{I}}\), is considered and shown to have strong implications for these questions. A major role is played by the (possibly sub-Riemannian) connections induced by stochastic differential equations: Damped Markovian connections are used for the covariant derivatives.

MSC:

60H07 Stochastic calculus of variations and the Malliavin calculus
58J65 Diffusion processes and stochastic analysis on manifolds
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Aida, S.: On the irreducibility of certain Dirichlet forms on loop spaces over compact homogeneous spaces. In: Elworthy, K.D., Kusuoka, S., Shigekawa, I. (eds.) New Trends in Stochastic Analysis. Proceedings of Taniguchi Symposium, Charingworth, 1995, pp. 3–42. World Scientific Press (1997)
[2] Aida, S., Elworthy, K.D.: Differential calculus on path and loop spaces. 1. Logarithmic Sobolev inequalities on path spaces. C. R. Acad. Sci. Paris, t. 321(série I):97–102 (1995) · Zbl 0837.60053
[3] Aida S. (1993). On the Ornstein–Uhlenbeck operators on Wiener–Riemannian manifolds. J. Funct. Anal. 116(1): 83–110 · Zbl 0795.60043 · doi:10.1006/jfan.1993.1105
[4] Aida S. (1995). Sobolev spaces over loop groups. J. Funct. Anal. 127(1): 155–172 · Zbl 0823.46034 · doi:10.1006/jfan.1995.1006
[5] Cruzeiro A.B., Fang S. (1995). Une inégalité l 2 pour des intégrales stochastiques anticipatives sur une variété riemannienne. C. R. Acad. Sci. Paris Sér I 321: 1245–1250 · Zbl 0845.60050
[6] Cruzeiro A.B., Malliavin P. (1996). Renormalized differrential geometry on path spaces: structural equation, curvature. J. Funct. Anal. 139: 119–181 · Zbl 0869.60060 · doi:10.1006/jfan.1996.0081
[7] Cruzeiro A.B., Malliavin P. (2000). Frame bundle of Riemannian path space and Ricci tensor in adapted differential geometry. J. Funct. Anal. 177: 219–253 · Zbl 1010.58026 · doi:10.1006/jfan.2000.3634
[8] Giuseppe Da Prato and Jerzy Zabczyk. Second order partial differential equations in Hilbert spaces, vol. 293 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (2002) · Zbl 1012.35001
[9] Driver B.K. (1992). A Cameron–Martin type quasi-invariance theorem for Brownian motion on a compact Riemannian manifold. J. Funct. Anal. 100: 272–377 · Zbl 0765.60064 · doi:10.1016/0022-1236(92)90035-H
[10] Driver B.K., Lohrenz T. (1996). Logarithmic Sobolev inequalities for pinned loop groups. J. Funct. Anal. 140(2): 381–448 · Zbl 0859.22012 · doi:10.1006/jfan.1996.0113
[11] Driver, B.K.: The non-equivalence of Dirichlet forms on path spaces. In: Stochastic analysis on infinite-dimensional spaces (Baton Rouge, LA, 1994), pp. 75–87. Longman Sci. Tech., Harlow (1994) · Zbl 0819.31005
[12] Eberle, A.: Uniqueness and non-uniqueness of semigroups generated by singular diffusion operators. Lecture Notes in Mathematics, vol. 1718. Springer, Heidelberg (1999) · Zbl 0957.60002
[13] Eells, J.: On the geometry of function spaces. In: Symposium Internacional de topologí a Algebraica International Symposium on Algebraic Topology, pp. 303–308. Universidad Nacional Autónoma de México and UNESCO, Mexico City (1958) · Zbl 0092.11302
[14] Eliasson H. (1967). Geometry of manifolds of maps. J. Diff. Geom. 1: 169–194 · Zbl 0163.43901
[15] Elworthy, K.D., LeJan, Y., Li, X.-M.: On the geometry of diffusion operators and stochastic flows, Lecture Notes in Mathematics, vol. 1720. Springer, Heidelberg (1999)
[16] Elworthy K.D., LeJan Y., Li X.-M. (1996). Integration by parts formulae for degenerate diffusion measures on path spaces and diffeomorphism groups. C.R. Acad. Sci. Paris t. 323(serié 1): 921–926 · Zbl 0865.60046
[17] Elworthy, K.D., LeJan, Y., Li, X.-M.: Concerning the geometry of stochastic differential equations and stochastic flows. In: Elworthy, K.D., Kusuoka, S., Shigekawa, I. (eds.) New Trends in Stochastic Analysis. Proceedings of Taniguchi Symposium, 1995, Charingworth. World Scientific Press (1997)
[18] Elworthy, K.D., Li, X.-M.: A class of integration by parts formulae in stochastic analysis I. In Itô’s Stochastic Calculus and Probability Theory (dedicated to Kiyosi Itô on the occasion of his eightieth birthday). Springer, Heidelberg (1996)
[19] Elworthy, K.D., Li, X.-M.: Special Itô maps and an L 2 Hodge theory for one forms on path spaces. In: Stochastic processes, physics and geometry: new interplays, I (Leipzig, 1999), pp. 145–162. Am. Math. Soc. (2000)
[20] Elworthy K.D., Li X.-M. (2003). Some families of q-vector fields on path spaces. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6(suppl.): 1–27 · Zbl 1073.58012 · doi:10.1142/S0219025703001213
[21] Elworthy, K.D., Li, X.-M.: Intertwining and the Markov uniqueness problem on path spaces. To appear in Stochastic Partial Differential Equations and Applications Đ VIIÓ, Lecture Notes in Pure and Applied Mathematics (2004)
[22] Elworthy K.D., Ma Z.-M. (1997). Admissible vector fields and related diffusions on finite-dimensional manifolds. Ukraïn. Mat. Zh. 49(3): 410–423 · Zbl 0941.58023
[23] Elworthy, K.D., Yor, M.: Conditional expectations for derivatives of certain stochastic flows. In: Azéma, J., Meyer, P.A., Yor, M. (eds.) Sem. de Prob. XXVII. Lecture Notes in Mathematics, vol. 1557, pp. 159–172. Springer, Heidelberg (1993) · Zbl 0795.60046
[24] Elworthy K.D., Li X.-M. (1998). Bismut type formulae for differential forms. C. R. Acad. Sci., Sér/ I Math. Paris 327(1): 87–92 · Zbl 1008.58026
[25] Elworthy K.D., Li X.-M. (2003). Gross-Sobolev spaces on path manifolds: uniqueness and intertwining by Itô maps. C. R. Math. Acad. Sci. Paris 337(11): 741–744 · Zbl 1037.58023
[26] Elworthy K.D., Ma Z.-M. (1997). Vector fields on mapping spaces and related Dirichlet forms and diffusions. Osaka J. Math. 34(3): 629–651 · Zbl 0893.31009
[27] Fang, S., Franchi, J.: A differentiable isomorphism between Wiener space and path group. In: Séminaire de Probabilités, XXXI, vol. 1655 of Lecture Notes in Math., pp. 54–61. Springer, Berlin (1997) · Zbl 0889.58084
[28] Freed D.S. (1988). The geometry of loop groups. J. Diff. Geom. 28(2): 223–276 · Zbl 0619.58003
[29] Hsu E.P. (1995). Quasi-invariance of the Wiener measure on the path space over a compact Riemannian manifold. J. Funct. Anal. 134(2): 417–450 · Zbl 0847.58082 · doi:10.1006/jfan.1995.1152
[30] Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, 2nd edn. North-Holland, Amsterdam (1989) · Zbl 0684.60040
[31] Jones, J.D.S., Léandre, R.: L p -Chen forms on loop spaces. In: Stochastic analysis (Durham, 1990), vol. 167 of London Math. Soc. Lecture Note Ser., pp. 103–162. Cambridge University Press, Cambridge (1991)
[32] Krée M., Krée P. (1983). Continuité de la divergence dans les espaces de sobolev relatifs éa l’espace de wiener. C. R. Acad. Sci. Paris Sèr. I Math 296(20): 833–836 · Zbl 0531.46034
[33] Léandre, R.: Integration by parts formulas and rotationally invariant Sobolev calculus on free loop spaces. J. Geom. Phys., 11(1–4):517–528 Infinite-dimensional geometry in physics (Karpacz, 1992) (1993)
[34] Li X.-D. (2003). Sobolev spaces and capacities theory on path spaces over a compact Riemannian manifold. Probab. Theory Relat. Fields 125: 96–134 · Zbl 1018.58026 · doi:10.1007/s004400200227
[35] Malliavin, P.: Stochastic analysis, Grundlehren der Mathematischen Wissenschaften, vol. 313. Springer, Heidelberg (1997) · Zbl 0878.60001
[36] Malliavin, P.: Stochastic calculus of variation and hypoelliptic operators. In: Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976), pp. 195–263. Wiley, New York (1978)
[37] Narasimhan M.S., Ramanan S. (1961). Existence of universal connections. Am. J. Math. 83: 563–572 · Zbl 0114.38203 · doi:10.2307/2372896
[38] Nualart D. (1995). The Malliavin Calculus and Related Topics. Springer, Heidelberg · Zbl 0837.60050
[39] Pardoux É., Peng S.G. (1990). Adapted solution of a backward stochastic differential equation. Syst Control Lett 14(1): 55–61 · Zbl 0692.93064 · doi:10.1016/0167-6911(90)90082-6
[40] Quillen D. (1988). Superconnections; character forms and the Cayley transform. Topology 27(2): 211–238 · Zbl 0671.57013 · doi:10.1016/0040-9383(88)90040-7
[41] Shigekawa I. (1995). A quasihomeomorphism on the Wiener space. Proc. Sympos. Pure math. Am. Math. Soc. 57: 473–486 · Zbl 0821.60059
[42] Shigekawa I. (1986). de Rham-Hodge-Kodaira’s decomposition on an abstract Wiener space. J. Math. Kyoto Univ. 26(2): 191–202 · Zbl 0611.58006
[43] Shigekawa, I.: Differential calculus on a based loop group. In: New trends in stochastic analysis (Charingworth, 1994), pages 375–398. World Sci. Publishing, River Edge (1997)
[44] Shigekawa, I., Taniguchi, S.: A Kähler metric on a based loop group and a covariant differentiation. In: Itô’s stochastic calculus and probability theory, pp. 327–346. Springer, Tokyo (1996) · Zbl 0867.53029
[45] Sugita H. (1985). On a characterization of the Sobolev spaces over an abstract wiener space. J. Math. Kyoto Univ. 25(4): 717–725 · Zbl 0614.46028
[46] Takeda M. (1985). On the uniqueness of Markovian self-adjoint extension of diffusion operators on infinite dimensional spaces. Osaka J. Math. 22(4): 733–742 · Zbl 0611.60074
[47] Yosida, K.: Functional analysis. Die Grundlehren der Mathematischen Wissenschaften, Band 123. Academic, New York (1965)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.