×

Asymptotics of spectral gaps on loop spaces over a class of Riemannian manifolds. (English) Zbl 1349.58016

The question of the existence of a spectral gap for the Ornstein-Uhlenbeck operator on the loop space over a Riemannian manifold \(M\) is altogether important in stochastic analysis, and very delicate. A. Eberle [J. Math. Pures Appl. (9) 81, No. 10, 915–955 (2002; Zbl 1029.58026)] provided a nice compact counterexample \(M\). No other generic answer is known regarding classes of compact Riemannian manifolds, and not even an example in the reverse direction.
The author addresses that question in the case in which the manifold \(M\) is rotationally symmetric about a pole \(y_0\). The particular subcase of hyperbolic spaces was already positively handled by X. Chen et al. [J. Funct. Anal. 259, No. 6, 1421–1442 (2010; Zbl 1228.58017)]. The author gives an alternative proof of the latter, which remains valid in the more generic case he considers (under some assumption on the given rotationally invariant metric).
Moreover, he then addresses the supplementary question of the asymptotic behaviour of the spectral gap \(g(\lambda)\) he just obtained, when the parameter \(\lambda\) goes to infinity, after having weighted the pinned Wiener measure by prescribing that the duration of paths running from some (arbitrary) fixed \(x_0\) to the pole \(y_0\) be equal to \(1/\lambda\). Namely, he establishes that \(g(\lambda)/\lambda\) goes to the bottom of the spectrum of the Hessian of the \(H^1\)-energy at the geodesic \(g(x_0,y_0)\) from \(x_0\) to \(y_0\).
Following the strategy that S.-Z. Fang [C. R. Acad. Sci., Paris, Sér. I 318, No. 3, 257–260 (1994; Zbl 0805.60056)] used in the simpler case of the path space over \(M\), the author’s fine method relies first on the Clark-Ocone-Haussmann formula and on the logarithmic Sobolev inequality. A major difficulty in the loop space setting comes from the singular drift of the Brownian bridge, which is not easy to handle. To overcome this difficulty, the author proceeds by a careful analysis of Jacobi fields about the geodesic \(g(x_0,y_0)\), and the use of rough paths (to get continuity at the limit) and quasi-sure analysis (to overcome the singularity of the pinned Wiener measure with respect to the Wiener measure).

MSC:

58J65 Diffusion processes and stochastic analysis on manifolds
47A11 Local spectral properties of linear operators
60H07 Stochastic calculus of variations and the Malliavin calculus
55P35 Loop spaces
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Aida, S., On the irreducibility of certain Dirichlet forms on loop spaces over compact homogeneous spaces, (New Trends in Stochastic Analysis. New Trends in Stochastic Analysis, Charingworth, 1994 (1997), World Sci. Publ.: World Sci. Publ. River Edge, NJ), 3-42
[2] Aida, S., Gradient estimates of harmonic functions and the asymptotics of spectral gaps on path spaces, Interdiscip. Inform. Sci., 2, 1, 75-84 (1996) · Zbl 0921.31007
[3] Aida, S., Logarithmic derivatives of heat kernels and logarithmic Sobolev inequalities with unbounded diffusion coefficients on loop spaces, J. Funct. Anal., 174, 2, 430-477 (2000) · Zbl 0968.58026
[4] Aida, S., Semiclassical limit of the lowest eigenvalue of a Schrödinger operator on a Wiener space, J. Funct. Anal., 203, 2, 401-424 (2003) · Zbl 1038.81027
[5] Aida, S., Precise Gaussian estimates of heat kernels on asymptotically flat Riemannian manifolds with poles, (Recent Developments in Stochastic Analysis and Related Topics, Proceedings of the First Sino-German Conference on Stochastic Analysis (2004)), 1-19 · Zbl 1160.58309
[6] Aida, S., Semi-classical limit of the bottom of spectrum of a Schrödinger operator on a path space over a compact Riemannian manifold, J. Funct. Anal., 251, 1, 59-121 (2007) · Zbl 1127.58014
[7] Aida, S., COH Formula and Dirichlet Laplacians on Small Domains of Pinned Path Spaces, Contemp. Math., vol. 545, 1-12 (2011), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 1237.58030
[8] Aida, S., Vanishing of one dimensional \(L^2\)-cohomologies of loop groups, J. Funct. Anal., 261, 8, 2164-2213 (2011) · Zbl 1236.58020
[9] Andersson, L.; Driver, B., Finite-dimensional approximations to Wiener measure and path integral formulas on manifolds, J. Funct. Anal., 165, 2, 430-498 (1999) · Zbl 0943.58024
[10] Besse, A., Einstein Manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 10 (1987), Springer-Verlag: Springer-Verlag Berlin · Zbl 0613.53001
[11] Capitaine, M.; Hsu, E.; Ledoux, M., Martingale representation and a simple proof of logarithmic Sobolev inequalities on path spaces, Electron. Commun. Probab., 2, 71-81 (1997) · Zbl 0890.60045
[12] Cattiaux, P.; Gentil, I.; Guillin, A., Weak logarithmic Sobolev inequalities and entropic convergence, Probab. Theory Related Fields, 139, 3-4, 563-603 (2007) · Zbl 1130.26010
[13] Chen, X.; Li, X.-M.; Wu, B., A Poincaré inequality on loop spaces, J. Funct. Anal., 259, 6, 1421-1442 (2010) · Zbl 1228.58017
[14] Chen, X.; Li, X.-M.; Wu, B., A spectral gap for the Brownian bridge measure on hyperbolic spaces, (Progress in Analysis and Its Applications (2010), World Sci. Publ.), 398-404 · Zbl 1271.58018
[15] Chen, X.; Li, X.-M.; Wu, B., A concrete estimate for the weak Poincaré inequality on loop space, Probab. Theory Related Fields, 151, 3-4, 559-590 (2011) · Zbl 1245.58017
[16] Chow, B.; S-C Chu, etal, The Ricci Flow: Techniques and Applications. Part I. Geometric Aspects, Mathematical Surveys and Monographs, vol. 135 (2007), American Mathematical Society: American Mathematical Society Providence, RI
[17] Cruzeiro, A. B.; Malliavin, P., Renormalized differential geometry on path space: structural equation, curvature, J. Funct. Anal., 139, 1, 119-181 (1996) · Zbl 0869.60060
[18] Driver, B. K., A Cameron-Martin type quasi-invariance theorem for Brownian motion on a compact Riemannian manifold, J. Funct. Anal., 110, 2, 272-376 (1992) · Zbl 0765.60064
[19] Driver, B. K., A Cameron-Martin type quasi-invariance theorem for pinned Brownian motion on a compact Riemannian manifold, Trans. Amer. Math. Soc., 342, 1, 375-395 (1994) · Zbl 0792.60013
[20] Driver, B. K., The non-equivalence of Dirichlet forms on path spaces, (Stochastic Analysis on Infinite-Dimensional Spaces. Stochastic Analysis on Infinite-Dimensional Spaces, Baton Rouge, LA, 1994. Stochastic Analysis on Infinite-Dimensional Spaces. Stochastic Analysis on Infinite-Dimensional Spaces, Baton Rouge, LA, 1994, Pitman Res. Notes Math. Ser., vol. 310 (1994), Longman Sci. Tech.: Longman Sci. Tech. Harlow), 75-87 · Zbl 0819.31005
[21] Eberle, A., Absence of spectral gaps on a class of loop spaces, J. Math. Pures Appl. (9), 81, 10, 915-955 (2002) · Zbl 1029.58026
[22] Eberle, A., Spectral gaps on discretized loop spaces, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 6, 2, 265-300 (2003) · Zbl 1056.58011
[23] Eberle, A., Local spectral gaps on loop spaces, J. Math. Pures Appl. (9), 82, 3, 313-365 (2003) · Zbl 1037.58022
[24] Elworthy, K. D.; Li, Xue-Mei, Itô maps and analysis on path spaces, Math. Z., 257, 3, 643-706 (2007) · Zbl 1181.60078
[25] Enchev, O.; Stroock, D. W., Integration by parts for pinned Brownian motion, Math. Res. Lett., 2, 2, 161-169 (1995) · Zbl 0845.58055
[26] Enchev, O.; Stroock, D. W., Pinned Brownian motion and its perturbations, Adv. Math., 119, 2, 127-154 (1996) · Zbl 0853.58111
[27] Fang, S., Inégalité du type de Poincaré sur l’espace des chemins riemanniens, C. R. Acad. Sci. Paris Sér. I Math., 318, 3, 257-260 (1994) · Zbl 0805.60056
[28] Friz, P.; Hairer, M., A Course on Rough Paths. With an Introduction to Regularity Structures, Universitext (2014), Springer · Zbl 1327.60013
[29] Friz, P.; Victoir, N., Multidimensional stochastic processes as rough paths, (Theory and Applications. Theory and Applications, Cambridge Studies in Advanced Mathematics, vol. 120 (2010), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 1193.60053
[30] Gong, F.; Ma, Z., The log-Sobolev inequality on loop space over a compact Riemannian manifold, J. Funct. Anal., 157, 2, 599-623 (1998) · Zbl 0909.22036
[31] Gordina, M., Quasi-invariance for the pinned Brownian motion on a Lie group, Stochastic Process. Appl., 104, 243-257 (2003) · Zbl 1075.58020
[32] Greene, R. E.; Wu, H., Function Theory on Manifolds Which Possess a Pole, Lecture Notes in Mathematics, vol. 699 (1979), Springer: Springer Berlin · Zbl 0414.53043
[33] Gross, L., Logarithmic Sobolev inequalities, Amer. J. Math., 97, 4, 1061-1083 (1975) · Zbl 0318.46049
[34] Helffer, B.; Nier, F., Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary, Mém. Soc. Math. Fr. (N.S.), 105 (2006) · Zbl 1108.58018
[35] Holley, R.; Kusuoka, S.; Stroock, D. W., Asymptotics of the spectral gap with applications to the theory of simulated annealing, J. Funct. Anal., 83, 2, 333-347 (1989) · Zbl 0706.58075
[36] Hsu, E., Stochastic Analysis on Manifolds, Graduate Studies in Mathematics, vol. 38 (2002), American Mathematical Society: American Mathematical Society Providence, RI
[37] Hsu, E., Quasi-invariance of the Wiener measure on path spaces: noncompact case, J. Funct. Anal., 193, 2, 278-290 (2002) · Zbl 1015.58014
[38] Ikeda, N.; Watanabe, S., Stochastic Differential Equations and Diffusion Processes, North-Holland Mathematical Library, vol. 24 (1989) · Zbl 0684.60040
[39] Inahama, Y., Large deviation principle of Freidlin-Wentzell type for pinned diffusion processes, Trans. Amer. Math. Soc., 367, 11, 8107-8137 (2015) · Zbl 1338.60083
[40] Jost, J., Riemannian Geometry and Geometric Analysis (1998), Springer · Zbl 0997.53500
[41] Laetsch, T., An approximation to Wiener measure and quantization of the Hamiltonian on manifolds with non-positive sectional curvature, J. Funct. Anal., 265, 8, 1667-1727 (2013) · Zbl 1298.81138
[42] Léandre, R., Integration by parts formulas and rotationally invariant Sobolev calculus on free loop spaces, J. Geom. Phys., 11, 1-4, 517-528 (1993) · Zbl 0786.60074
[43] Ledoux, M.; Qian, Z.; Zhang, T., Large deviations and support theorem for diffusions via rough paths, Stochastic Process. Appl., 102, 2, 265-283 (2002) · Zbl 1075.60510
[44] Li, P.; Yau, S. T., On the parabolic kernel of the Schrödinger operator, Acta Math., 156, 3-4, 153-201 (1986)
[45] Lim, A., Path integrals on a compact manifold with non-negative curvature, Rev. Math. Phys., 19, 9, 967-1044 (2007) · Zbl 1148.58005
[46] Lyons, T., Differential equations driven by rough signals, Rev. Mat. Iberoam., 14, 215-310 (1998) · Zbl 0923.34056
[47] Lyons, T.; Caruana, M.; Lévy, T., Differential equations driven by rough paths, (Ecole d’Eté de Probabilités de Saint-Flour XXXIV-2004. Ecole d’Eté de Probabilités de Saint-Flour XXXIV-2004, Lecture Notes in Mathematics, vol. 1908 (2007), Springer-Verlag: Springer-Verlag Berlin, Heidelberg)
[48] Lyons, T.; Qian, Z., System Control and Rough Paths, Oxford Mathematical Monographs (2002) · Zbl 1029.93001
[49] Malliavin, P.; Stroock, D. W., Short time behavior of the heat kernel and its logarithmic derivatives, J. Differential Geom., 44, 3, 550-570 (1996) · Zbl 0873.58063
[50] Nualart, D., The Malliavin Calculus and Related Topics, Probability and Its Applications (2006), Springer-Verlag: Springer-Verlag Berlin · Zbl 1099.60003
[51] Sasamori, T., On estimates of heat kernels on Riemannian manifolds with poles (March 2015), Master Thesis
[52] Simon, B., Semiclassical analysis of low lying eigenvalues I. Nondegenerate minima: asymptotic expansions, Ann. Inst. Henri Poincaré, X, 4, 295-308 (1983) · Zbl 0526.35027
[53] Shigekawa, I., Stochastic Analysis, Iwanami Series in Modern Mathematics (2004), American Mathematical Society: American Mathematical Society Providence, RI, Translations of Mathematical Monographs, vol. 224
[54] Stroock, D. W., An estimate on the Hessian of the heat kernel, (Itô’s Stochastic Calculus and Probability Theory (1996), Springer: Springer Tokyo), 355-371 · Zbl 0868.58075
[55] Watanabe, S., Analysis of Wiener functionals (Malliavin calculus) and its applications to heat kernels, Ann. Probab., 15, 1, 1-39 (1987) · Zbl 0633.60077
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.