Quasi-invariant measures and their characterization by conditional probabilities

Based on quasi-invariance properties of the Gaussian process and the gamma process, we give certain formal expressions of their laws on infinite-dimensional spaces. They are also consistent with some identities of conditional probabilities which are shown to be equivalent to the quasi-invariance properties.     

Quasi-invariant measures on the infinite product ofSU(2)

Investigated is quasi-invariance of power probabilities on the infinite product ofSU(2). We consider the subgroup consisting of those actions which keep a measure quasi-invariant (i.e., mutually absolutely continuous) and call it the quasi-invariant subgroup of the measure. We establish several estimations for the quasi-invariant subgroups in terms oflf_p-type subgroups ofSU(2). Our methods are based on Hellinger integrals, Fourier analysis, and spherical functions onSU(2).     

Quasi-invariant domain constants

Five domain constants are studied in our paper, all related to the hyperbolic geometry in hyperbolic plane regions which are uniformly perfect (in Pommerenke’s terminology). Relations among these domain constants are obtained, from which bounds are derived for the variance ratio of each constant under conformal mappings of the regions, and we also show that each constant may be used to characterize uniformly perfect regions. The authors wish to thank the University of California, San Diego for its hospitality during the 1988–89 academic year when this research was begun. Supported by the Landau Center for Mathematical Research in Analysis. Research partially supported by NSF Grant No. DMS-8801439.     

Quasi-invariant transformations on the path space over a manifold

Some interesting quasi-invariant transformations on the path space over a Riemannian manifold are investigated. The results improve some previous ones. An erratum to this article is available at .     

Quasi-invariant manifolds,stability, and generalized Hopf bifurcation

Sunto Viene compiuta un'analisi completa del problema della biforcazione di Hopf relativa ad arbitrarie piccole perturbazioni del secondo membro di un'equazione differenziale in Rⁿ, p=f₀(p). Gli autovalori di f₀(O) soddisfano una condizione di non risonanza. I risultati sono forniti in termini delle proprietá di stabilità di un sistema dinamico piano convenientemente associato all'equazione imperturbata.

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Research partially supported by U.S. Army Research Grant DAAG-29-80-C-0060 and by C.N.R. (Italian Council of Research) contr. 79.00696.01.

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Work performed under the auspices of the National Group of Math, Phys. of C.N.R.     

Some examples and the connection between cylindrical measures and measurable norms

In this paper, we construct an example of a cylindrical measure μ and a norm ‖·‖ such that ∥·∥ is μ-measurable by Dudley et al. and not μ-measurable by Gross and moreover is γ-measurable.     

Extensions of spaces with cylindrical measures and supports of measures determined by the Lévy Laplacian

Extensions of noncountably additive (cylindrical) measures are described, and examples of Hilbert supports of the Lévy-Gauss measure are given.Translated fromMatematicheskie Zametki, Vol. 64, No. 4, pp. 483–492, October, 1998.L. Accardi acknowledges partial support of the Russian Foundation for Basic Research under grant No. 96-01-00030.     

Quasi-invariant stochastic flows of SDEs with non-smooth drifts on compact manifolds

In this article we prove that stochastic differential equation (SDE) with Sobolev drift on a compact Riemannian manifold admits a unique ν-almost everywhere stochastic invertible flow, where ν is the Riemannian measure, which is quasi-invariant with respect to ν. In particular, we extend the well-known DiPerna-Lions flows of ODEs to SDEs on a Riemannian manifold.     

具拟不变测度群胚流上的Toeplitz代数

设G为第二可数群胚,具有Haar系{λ^n},R为实数群,左不变作用在G上。本文我们自然地（作为[3]中概念的推广）引进两类Toeplitz代数（对应于一拟不变测度）,即小和大Toeplitz代数,并且得到了小Toeplitz代数的同构定理及k群。     

Radon extension of cylindrical measures on weak dual multi-Hilbertian spaces and the generalized Bochner theorem

This paper presents a nonstandard approach to Radon extension of cylindrical measures on weak dual multi-Hilbertian spaces. The results obtained are applied to characterize the Fourier transforms of Radon measures on weak dual multi-Hilbertian spaces (the generalized Bochner theorem).     

Asymptotically cylindrical Ricci-flat manifolds

Asymptotically cylindrical Ricci-flat manifolds play a key role in constructing Topological Quantum Field Theories. It is particularly important to understand their behavior at the cylindrical ends and the natural restrictions on the geometry. In this paper we show that an orientable, connected, asymptotically cylindrical manifold with Ricci-flat metric can have at most two cylindrical ends. In the case where there are two such cylindrical ends, then there is reduction in the holonomy group Hol and is a cylinder.

     

Singular measures and Hausdorff measures

An example is given of a family of singular probability measures on the unit interval which are supported on a set of fractional Hausdorff dimension but cannot be represented as Hausdorff measures.     

Harmonic measures and bowen-margulis measures

We compare two families of measures defined on the absolute of the universal cover of a compact negatively-curved manifold: the harmonic measures and the Bowen-Margulis measures.     

Immersion ofn-dimensional cylindrical metrics in the form of cylindrical surfaces into Lobachevsky spaces

A definition of l -dimensional k-cylindrical metrics is introduced and it is proved that k-cylindrical metrics admit an immersion in the form of k-cylindrical surfaces into Lobachevsky space L^l+p, where codimension p of the immersion coincides with the codimension of the immersion of the base of the cylindrical metric into a Euclidean or Lobachevsky space.Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 66–69, 1991.     

Fuzzy measures and measures of fuzziness

In this paper, the fuzzy integral defined by Z.-X. Wang (Fuzzy Math. Wuhan, China, in press), which is different from that defined by M. Sugeno (“Theory of Fuzzy Integrals and Its Applications,” Ph. D., Tokyo Inst. of Technology, 1974), is further considered, and it is shown that the fuzzy measures of ordinary sets and fuzzy sets can be determined by each other. Summing up the results on the measure of fuzziness by A. DeLuca and S. A. Termini (Inform. and Control20 (1972), 301–312), Z.-X. Wang (op. cit.) and R. R. Yager (Internat. J. Gen. Systems5 (1979), 221–229; Inform. and Control44 (1980), 236–260), the axioms for measures of fuzziness are given. Furthermore, as an application of the furry integrals, a measure of fuzziness is defined. Inversely, it is proven that a measure of fuzziness satisfying some conditions can surely be expressed as a fuzzy integral with respect to some fuzzy measure.