×

zbMATH — the first resource for mathematics

Finite dimensional approximations to Wiener measure and path integral formulas on manifolds. (English) Zbl 0943.58024
The authors develop certain natural geometric approximation schemes for the Wiener measure on a compact Riemannian manifold. These approximations mimic the informal path integral formulas used for representing the heat semigroup on Riemannian manifolds. The approximation of the Wiener measure is made by measures on spaces of piecewise geodesics [see also M. A. Pinsky, Probabilistic analysis and related topics, Vol. 1, 199-236 (1978; Zbl 0452.60083) or R. W. R. Darling, Stochastics 12, 277-301 (1984; Zbl 0543.58028)]. The authors interpret \({\mathcal D}\sigma\) in the heuristic expression of the Wiener measure \(d\nu(\sigma)=\frac{1}{Z}e^{-(1/2)E(\sigma)}{\mathcal D}(\sigma)\), as a Riemannian volume form relative to a suitable metric. The authors use the Wong-Zakai type approximation theorem [E. Wong and M. Zakai, Int. J. Eng. Sci. 3, 213-229 (1965; Zbl 0131.16401)] for stochastic differential equations instead of weak convergence arguments. This fact allows them to get a stronger form of convergence which is needed in the proof of the integration by parts formula for the Wiener measure.

MSC:
58J65 Diffusion processes and stochastic analysis on manifolds
60J60 Diffusion processes
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aida, S., On the irreducibility of certain Dirichlet forms on loop spaces over compact homogeneous spaces, (), 3-42
[2] Airault, H.; Malliavin, P., Integration by parts formulas and dilatation vector fields on elliptic probability spaces, Prob. theory related fields, 106, 447-494, (1996) · Zbl 0867.60031
[3] Amit, Y., A multiflow approximation to diffusions, Stochastic process. appl., 37, 213-237, (1991) · Zbl 0734.60080
[4] Atiyah, M.F., Circular symmetry and stationary-phase approximation, Astérisque, 131, 43-59, (1985)
[5] Bally, V., Approximation for the solutions of stochastic differential equations. I. L^p-convergence, Stochastics stochastics rep., 28, 209-246, (1989) · Zbl 0695.60062
[6] Bally, V., Approximation for the solutions of stochastic differential equations. II. strong convergence, Stochastics stochastics rep., 28, 357-385, (1989) · Zbl 0695.60063
[7] Bell, D.R., The Malliavin calculus, Pitman monographs and surveys in pure and applied mathematics, (1987), Longman Harlow · Zbl 0678.60042
[8] Bell, D.R., Degenerate stochastic differential equations and hypoellipticity, Pitman monographs and surveys in pure and applied mathematics, (1995), Longman Harlow · Zbl 0859.60051
[9] Bismut, J.-M., Mecanique aleatoire, Lecture notes in mathematics, (1981), Springer-Verlag New York/Berlin · Zbl 0457.60002
[10] Bismut, J.-M., Large deviations and the Malliavin calculus, Progress in mathematics, (1984), Birkhäuser Boston · Zbl 0537.35003
[11] Bismut, J.-M., Index theorem and equivariant cohomology on the loop space, Comm. math. phys., 98, 213-237, (1985) · Zbl 0591.58027
[12] Blum, G., A note on the central limit theorem for geodesic random walks, Bull. austral. math. soc., 30, 169-173, (1984) · Zbl 0561.60071
[13] Cartan, É., Leçons sur la géometrie projective complexe. la théorie des groupes finis et continus et la géométrie différentielle traitées par la méthode du repère mobile; leçons sur la théorie des espaces à connexion projective, (1992), Gauthier-Villars Paris
[14] Chavel, I., Riemannian geometry—A modern introduction, Cambridge tracts in mathematics, (1993), Cambridge Press Cambridge · Zbl 0810.53001
[15] Cheng, K.S., Quantization of a general dynamical system by Feynman’s path integration formulation, J. math. phys., 15, 220-224, (1974)
[16] Chorin, A.J.; McCracken, M.F.; Hughes, T.J.R.; Marsden, J.E., Product formulas and numerical algorithms, Comm. pure appl. math., 31, 205-256, (1978) · Zbl 0358.65082
[17] Cruzeiro, A.-B.; Malliavin, P., Renormalized differential geometry on path space: structural equation, curvature, J. funct. anal., 139, 119-181, (1996) · Zbl 0869.60060
[18] Darling, R.W.R., On the convergence of gangolli processes to Brownian motion on a manifold, Stochastics, 12, 277-301, (1984) · Zbl 0543.58028
[19] de Boer, J.; Peeters, B.; Skenderis, K.; van Nieuwenhuizen, P., Loop calculations in quantum-mechanical non-linear sigma models, Nuclear phys. B, 446, 211-222, (1995) · Zbl 1009.81526
[20] DeWitt, B., Supermanifolds, Cambridge monographs on mathematical physics, (1992), Cambridge Univ. Press Cambridge
[21] Morette DeWitt, C., Feynman’s path integral, Comm. math. phys., 28, 47-67, (1972) · Zbl 0239.46041
[22] DeWitt-Morette, C.; Elworthy, K.D.; Nelson, B.L.; Sammelman, G.S., A stochastic scheme for constructing solutions of the Schrödinger equations, Ann. inst. H. Poincaré sect. A, 32, 327-341, (1980) · Zbl 0446.60045
[23] DeWitt-Morette, C., Path integrals in Riemannian manifolds, Lecture notes in phys., 39, (1975), Springer-Verlag New York/Berlin, p. 535-542
[24] DeWitt-Morette, C.; Elsworthy, K.D., New stochastic methods in physics, (1981), North-Holland Amsterdam
[25] DeWitt-Morette, C.; Maheshwari, Amar; Nelson, Bruce, Path integration in nonrelativistic quantum mechanics, Phys. rep., 50, 255-372, (1979)
[26] Dirac, P.A.M., Phys. Z. sowjetunion, 3, 64, (1933)
[27] Doss, H., Connections between stochastic and ordinary integral equations, Biological growth and spread (proc. conf., Heidelberg, 1979), Lecture notes in biomath., 38, (1979), Springer-Verlag New York/Berlin, p. 443-448
[28] Driver, B.K., Classifications of bundle connection pairs by parallel translation and lassos, J. funct. anal., 83, 185-231, (1989) · Zbl 0676.53033
[29] Driver, B.K., YM2: continuum expectations, lattice convergence, and lassos, Comm. math. phys., 123, 575-616, (1989) · Zbl 0819.58043
[30] Driver, B.K., A cameron – martin type quasi-invariance theorem for Brownian motion on a Riemannian manifold, J. funct. anal., 110, 272-376, (1992) · Zbl 0765.60064
[31] Driver, B.K., Integration by parts for heat kernel measures revisited, J. math. pures appl., 76, 703-737, (1997) · Zbl 0907.60052
[32] Driver, B.K., The Lie bracket of adapted vector fields on Wiener spaces, Appl. math. optim., 39, 179-210, (1999) · Zbl 0924.60032
[33] B. K. Driver, and, Y. Hu, Wong-zakai type approximation theorems, http://math.ucsd.edu/driver/, 1998.
[34] Eells, J.; Elworthy, K.D., Wiener integration on certain manifolds, Problems in non-linear analysis, C.I.M.E., IV ciclo, varenna, 1970, (1971), Edizioni Cremonese · Zbl 0226.58007
[35] J. Eells, Jr., On the geometry of function spaces, in Symposium internacional de topologı́a algebraica (International symposium on Algebraic Topology), Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958 pp. 303-308.
[36] Eliasson, H.I., Geometry of manifolds of maps, J. differential geom., 1, 169-194, (1967) · Zbl 0163.43901
[37] Elworthy, K.D., Gaussian measures on Banach spaces and manifolds, Global analysis and its applications (lectures, internat. sem. course, internat. centre theoret. phys., trieste, 1972), (1974), Internat. Atomic Energy Agency Vienna, p. 151-166 · Zbl 0319.58007
[38] Elworthy, K.D., Measures on infinite-dimensional manifolds, Functional integration and its applications, proc. internat. conf., London, 1974, (1975), Clarendon Oxford, p. 60-68 · Zbl 0342.58009
[39] Elworthy, K.D., Stochastic dynamical systems and their flows, Stochastic analysis, proc. internat. conf., northwestern university, evanston, IL, 1978, (1978), Academic Press New York/London, p. 79-95 · Zbl 0439.60065
[40] Elworthy, K.D.; Le Jan, Y.; Li, X.-M., Integration by parts formulae for degenerate diffusion measures on path spaces and diffeomorphisms groups, C. R. acad. sci. Paris Sér. I math., 323, 921-926, (1996) · Zbl 0865.60046
[41] Elworthy, K.D.; Li, X.-M., Formulae for the derivatives of heat semigroups, J. funct. anal., 125, 252-286, (1994) · Zbl 0813.60049
[42] Elworthy, K.D.; Li, X.M., A class of integration by parts formulae in stochastic analysis, I, Itô’s stochastic calculus and probability theory, (1996), Springer-Verlag Tokyo, p. 15-30 · Zbl 0881.60052
[43] Emery, M., Stochastic calculus in manifolds, (1989), Springer-VerlagUniversitext Berlin
[44] Enchev, O.; Stroock, D.W., Towards a riemannian geometry on the path space over a Riemannian manifold, J. funct. anal., 134, 392-416, (1995) · Zbl 0847.58080
[45] Ethier, S.N.; Kurtz, T.G., Markov processes, Wiley series in probability and mathematical statistics: probability and mathematical statistics, (1986), Wiley New York · Zbl 0592.60049
[46] Fang, S.Z.; Malliavin, P., Stochastic analysis on the path space of a riemannian manifold, I. Markovian stochastic calculus, J. funct. anal., 118, 249-274, (1993) · Zbl 0798.58080
[47] Feynman, R.P., Space-time approach to non-relativistic quantum mechanics, Rev. modern phys., 20, 367-387, (1948) · Zbl 1371.81126
[48] Flaschel, P.; Klingenberg, W., Riemannsche hilbertmannigfaltigkeiten, periodisch geodätischen, mit einem anhang von H. Karcher, Lecture notes in mathematics, (1972), Springer-Verlag Berlin · Zbl 0238.58009
[49] Fujiwara, D., Remarks on convergence of the Feynman path integrals, Duke math. J., 47, 559-600, (1980) · Zbl 0457.35026
[50] Fulling, S.A., Pseudodifferential operators, covariant quantization, the inescapable Van vleck – morette determinant, and the R/6 controversy, Int. J. modern phys. D, 5, 597-608, (1996)
[51] Gangolli, R., On the construction of certain diffusions on a differentiable manifold, Z. wahrsch. verw. gebiete, 2, 406-419, (1964) · Zbl 0132.12702
[52] Glimm, J.; Jaffe, A., Quantum physics, (1987), Springer-Verlag New York
[53] Gross, L., A Poincaré lemma for connection forms, J. funct. anal., 63, 1-46, (1985) · Zbl 0624.53021
[54] Gross, L., Lattice gauge theory; heuristics and convergence, Stochastic processes—mathematics and physics, bielefeld, 1984, Lecture notes in math., 1158, (1986), Springer-Verlag Berlin, p. 130-140
[55] Guo, S.J., On the mollifier approximation for solutions of stochastic differential equations, J. math. Kyoto univ., 22, 243-254, (1982) · Zbl 0496.60062
[56] Hsu, E.P., Flows and quasi-invariance of the Wiener measure on path spaces, Stochastic analysis, Ithaca, NY, 1993, Proc. sympos. pure math., 57, (1995), Amer. Math. Soc Providence, p. 265-279 · Zbl 0830.58035
[57] Hsu, E.P., Quasi-invariance of the Wiener measure on the path space over a compact Riemannian manifold, J. funct. anal., 134, 417-450, (1995) · Zbl 0847.58082
[58] Ichinose, W., On the formulation of the Feynman path integral through broken line paths, Comm. math. phys., 189, 17-33, (1997) · Zbl 0892.58012
[59] Ikeda, N.; Watanabe, S., Stochastic differential equations and diffusion processes, (1981), North-Holland Amsterdam · Zbl 0495.60005
[60] Inoue, A.; Maeda, Y., On integral transformations associated with a certain lagrangian—As a prototype of quantization, J. math. soc. Japan, 37, 219-244, (1985) · Zbl 0584.58019
[61] Jørgensen, E., The central limit problem for geodesic random walks, Z. wahrsch. verw. gebiete, 32, 1-64, (1975) · Zbl 0292.60103
[62] Kaneko, H.; Nakao, S., A note on approximation for stochastic differential equations, Séminaire de probabilités, XXII, Lecture notes in math., 1321, (1988), Springer-Verlag New York/Berlin, p. 155-162 · Zbl 0647.60072
[63] Karatzas, I.; Shreve, S.E., Brownian motion and stochastic calculus, Graduate texts in mathematics, 113, (1991), Springer-Verlag Berlin
[64] Klingenberg, W., Lectures on closed geodesics, Grundlehren der mathematischen wissenschaften, 230, (1978), Springer-Verlag Berlin
[65] Kobayashi, S., Theory of connections, Ann. mat. pura appl. (4), 43, 119-194, (1957) · Zbl 0124.37604
[66] Kunita, H., Stochastic flows and stochastic differential equations, Cambridge studies in advanced mathematics, (1990), Cambridge Univ. Press Cambridge · Zbl 0743.60052
[67] Kuo, H.Hsiung, Gaussian measures in Banach spaces, Lecture notes in mathematics, 463, (1975), Springer-Verlag Berlin
[68] Kurtz, T.G.; Protter, P., Weak limit theorems for stochastic integrals and stochastic differential equations, Ann. probab., 19, 1035-1070, (1991) · Zbl 0742.60053
[69] Kurtz, T.G.; Protter, P., Wong – zakai corrections, random evolutions, and simulation schemes for sdes, Stochastic analysis, (1991), Academic Press Boston, p. 331-346 · Zbl 0762.60047
[70] Léandre, R., Integration by parts formulas and rotationally invariant Sobolev calculus on free loop spaces, J. geom. phys., 11, 517-528, (1993) · Zbl 0786.60074
[71] R. Léandre, Bismut-Nualart-Pardoux cohomology and entire Hochschild cohomology, preprint, 1994.
[72] Léandre, R., Invariant Sobolev calculus on the free loop space, Acta appl. math., 46, 267-350, (1997) · Zbl 0895.60055
[73] Léandre, R.; Norris, J.R., Integration by parts and cameron – martin formulas for the free path space of a compact Riemannian manifold, Séminaire de probabilités, XXXI, Lecture notes in math., 1655, (1997), Springer-Verlag New York/Berlin, p. 16-23 · Zbl 0889.60058
[74] Lyons, T.J.; Qian, Z.M., Calculus for multiplicative functionals, Itô’s formula and differential equations, Stochastic calculus and probability theory, (1996), Springer-Verlag Tokyo, p. 233-250 · Zbl 0862.60043
[75] Lyons, T.; Qian, Z., Stochastic Jacobi fields and vector fields induced by varying area on path spaces, Probab. theory related fields, 109, 539-570, (1997) · Zbl 0903.60008
[76] Malliavin, P., Geometrie differentielle stochastique, Séminaire de mathématiques supérieures, (1978), Presses Univ. Montréal Montreal
[77] Malliavin, P., Stochastic calculus of variation and hypoelliptic operators, Proceedings of the international symposium on stochastic differential equations, res. inst. math. sci., Kyoto univ., Kyoto, 1976, (1978), Wiley New York, p. 195-263
[78] Milliavin, P., Stochastic Jacobi fields, Partial differential equations and geometry, proc. conf., park city, ut, 1977, Lecture notes in pure and appl. math., 48, (1979), Dekker New York, p. 203-235
[79] Malliavin, P., Stochastic analysis, Grundlehren der mathematischen wissenschaften, 313, (1997), Springer-Verlag Berlin
[80] McShane, E.J., Stochastic differential equations and models of random processes, Proceedings of the sixth Berkeley symposium on mathematical statistics and probability univ. California, Berkeley, 1970/1971, (1972), Univ. of California Press Berkeley, p. 263-294 · Zbl 0283.60061
[81] McShane, E.J., Stochastic calculus and stochastic models, Probability and mathematical statistics, 25, (1974), Academic Press New York · Zbl 0292.60090
[82] Moulinier, J.-M., Théorème pour LES équations différentielles stochastiques, Bull. sci. math. (2), 112, 185-209, (1988) · Zbl 0655.60047
[83] Nakao, S.; Yamato, Y., Approximation theorem on stochastic differential equations, Proceedings of the international symposium on stochastic differential equations, res. inst. math. sci., Kyoto univ., Kyoto, 1976, (1978), Wiley New York, p. 283-296
[84] Norris, J.R., Twisted sheets, J. funct. anal., 132, 273-334, (1995) · Zbl 0848.60055
[85] Palais, R.S., Morse theory on Hilbert manifolds, Topology, 2, 299-340, (1963) · Zbl 0122.10702
[86] Pinsky, M.A., Stochastic Riemannian geometry, Probabilistic analysis and related topics, (1978), Academic Press New York, p. 199-236
[87] Revuz, D.; Yor, M., Continuous maringales and Brownian motion, Grundlehren der mathematischen wissenschaften, 293, (1994), Springer-Verlag Berlin
[88] Stroock, D.; Taniguchi, S., Diffusions as integral curves, or Stratonovich without Itô, The Dynkin festschrift, Progr. probab., 34, (1994), Birkhäuser Boston, p. 333-369 · Zbl 0814.60074
[89] Stroock, D.W.; Varadhan, S.R.S., On the support of diffusion processes with applications to the strong maximum principle, Proceedings of the sixth Berkeley symposium on mathematical statistics and probability, university California, Berkeley, 1970/1971, (1972), Univ. of California Press Berkeley, p. 333-359 · Zbl 0255.60056
[90] Stroock, D.W.; Varadhan, S.R.S., Diffusion processes with continuous coefficients, II, Comm. pure appl. math., 22, 479-530, (1969) · Zbl 0175.44802
[91] Stroock, D.W.; Srinivasa Varadhan, S.R., Multidimensional diffusion processes, Grundlehren der mathematischen wissenschaften, 233, (1979), Springer-Verlag Berlin · Zbl 0426.60069
[92] Sussmann, H.J., Limits of the wong – zakai type with a modified drift term, Stochastic analysis, (1991), Academic Press Boston, p. 475-493
[93] Takahashi, Y.; Watanabe, S., The probability functionals (onsager – machlup functions) of diffusion processes, Stochastic integrals, proc. sympos., univ. Durham, 1980, Lecture notes in math., 851, (1981), Springer-Verlag Berlin, p. 433-463 · Zbl 0625.60058
[94] Um, G.S., On normalization problems of the path integral method, J. math. phys., 15, 220-224, (1974)
[95] Wong, E.; Zakai, M., On the relation between ordinary and stochastic differential equations, Int. J. eng. sci., 3, 213-229, (1965) · Zbl 0131.16401
[96] Wong, E.; Zakai, M., On the relation between ordinary and stochastic differential equations and applications to stochastic problems in control theory, Automatic and remote control, III, proc. third congr. internat. fed. autom. control, IFAC, London, 1966, (1967), Inst. Mech. Engrs London, p. 5-
[97] Woodhouse, N.M.J., Geometric quantization, Oxford mathematical monographs, (1992), Clarendon Oxford · Zbl 0747.58004
[98] Wu, Jyh-Yang, Tech. rep., (November 1998)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.