Asymptotic behavior of divergences and Cameron-Martin theorem on loop spaces

We first prove the L^p-convergence (p\geq 1) and a Fernique-type exponential integrability of divergence functionals for all Cameron-Martin vector fields with respect to the pinned Wiener measure on loop spaces over a compact Riemannian manifold. We then prove that the Driver flow is a smooth transform on path spaces in the sense of the Malliavin calculus and has an \infty-quasi-continuous modification which can be quasi-surely well defined on path spaces. This leads us to construct the Driver flow on loop spaces through the corresponding flow on path spaces. Combining these two results with the Cruzeiro lemma [J. Funct. Anal. 54 (1983) 206-227] we give an alternative proof of the quasi-invariance of the pinned Wiener measure under Driver's flow on loop spaces which was established earlier by Driver [Trans. Amer. Math. Soc. 342 (1994) 375-394] and Enchev and Stroock [Adv. Math. 119 (1996) 127-154] by Doob's h-processes approach together with the short time estimates of the gradient and the Hessian of the logarithmic heat kernel on compact Riemannian manifolds. We also establish the L^p-convergence (p\geq 1) and a Fernique-type exponential integrability theorem for the stochastic anti-development of pinned Brownian motions on compact Riemannian manifold with an explicit exponential exponent. Our results generalize and sharpen some earlier results due to Gross [J. Funct. Anal. 102 (1991) 268-313] and Hsu [Math. Ann. 309 (1997) 331-339]. Our method does not need any heat kernel estimate and is based on quasi-sure analysis and Sobolev estimates on path spaces.Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000004

Spatial Reflection and Associated String Order in Quantum Spin Chains

We investigate spatial reflection and associated nonlocal order in spin chain quantum systems. The proposed string order parameters, e.g., reflected via operations of the spatial reflection or combinations of it with spin reflection, are able to characterize a variety of physical systems and allow us to gain renewed insights to the statistical mechanism underlying phenomena such as the Haldane gap and quantum phase transitions. Besides revealing further the potential application of the generalized parity symmetry in numerical algorithm, we build an explicit scheme to determine the symmetry and the related string order for matrix product states so that one can construct ansatz models with presumed properties.Comment: 4 pages, 1 figure, version accepted for publication in Phys. Rev.

Finding Transition Pathways on Manifolds

We consider noise-induced transition paths in randomly perturbed dynami- cal systems on a smooth manifold. The classical Freidlin-Wentzell large devia- tion theory in Euclidean spaces is generalized and new forms of action functionals are derived in the spaces of functions and the space of curves to accommodate the intrinsic constraints associated with the manifold. Numerical meth- ods are proposed to compute the minimum action paths for the systems with constraints. The examples of conformational transition paths for a single and double rod molecules arising in polymer science are numerically investigated

A parametrized compactness theorem under bounded Ricci curvature

We prove a parametrized compactness theorem on manifolds of bounded Ricci curvature, upper bounded diameter and lower bounded injectivity radius.Comment: 17 pages. Final version to appear in Front. Math. China. Reformulation of Theorem B to Corollary 1, adding some remarks, changing the precompactness in Corollary 1.3 (now Corollary 2) to $C^{0,\alpha}$-norm, and correcting some typo