Shy and Fixed-Distance Couplings of Brownian Motions on Manifolds

In this paper we introduce three Markovian couplings of Brownian motions on smooth Riemannian manifolds without boundary which sit at the crossroad of two concepts. The first concept is the one of shy coupling put forward in \cite{Burdzy-Benjamini} and the second concept is the lower bound on the Ricci curvature and the connection with couplings made in \cite{ReSt}. The first construction is the shy coupling, the second one is a fixed-distance coupling and the third is a coupling in which the distance between the processes is a deterministic exponential function of time. The result proved here is that an arbitrary Riemannian manifold satisfying some technical conditions supports shy couplings. If in addition, the Ricci curvature is non-negative, there exist fixed-distance couplings. Furthermore, if the Ricci curvature is bounded below by a positive constant, then there exists a coupling of Brownian motions for which the distance between the processes is a decreasing exponential function of time. The constructions use the intrinsic geometry, and relies on an extension of the notion of frames which plays an important role for even dimensional manifolds. In fact, we provide a wider class of couplings in which the distance function is deterministic in Theorem \ref{t:100} and Corollary~\ref{Cor:9}. As an application of the fixed-distance coupling we derive a maximum principle for the gradient of harmonic functions on manifolds with non-negative Ricci curvature. As far as we are aware of, these constructions are new, though the existence of shy couplings on manifolds is suggested by Kendall in \cite{Kendall}.Comment: This version is a refinement expansion and simplification of the previous versio

Metallic multilayers for X-band Bragg reflector applications

We present a structural and high frequency (8.72GHz) electrical characterization of sputter deposited Ti/W, Ti/Ru and Mo/Ti metallic multilayers for potential application as acoustic Bragg reflectors. We prove that all metallic multilayers comprised of different acoustic impedance metals such as Ti, W, Mo are promising candidates for Bragg reflector/bottom electrode in full X-band thin film acoustic resonators. Values for high frequency resistivity of the order of $10^{-8} ohm.m$ are measured by use of a contact-free/non-invasive sheet resistance method

The Dirac field in Taub-NUT background

We investigate the SO(4,1) gauge-invariant theory of the Dirac fermions in the external field of the Kaluza-Klein monopole, pointing out that the quantum modes can be recovered from a Klein-Gordon equation analogous to the Schr\" odinger equation in the Taub-NUT background. Moreover, we show that there is a large collection of observables that can be directly derived from those of the scalar theory. These offer many possibilities of choosing complete sets of commuting operators which determine the quantum modes. In addition there are some spin- like and Dirac-type operators involving the covariantly constant Killing-Yano tensors of the hyper-K\" ahler Taub-NUT space. The energy eigenspinors of the central modes in spherical coordinates are completely evaluated in explicit, closed form.Comment: 20 pages, latex, no figure