Showing papers on "Gaussian measure published in 1993"

Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality

Abstract: where de denotes normalized surface measure, V is the conformal gradient and q = (2n)/(n 2). A modern folklore theorem is that by taking the infinitedimensional limit of this inequality, one obtains the Gross logarithmic Sobolev inequality for Gaussian measure, which determines Nelson's hypercontractive estimates for the Hermite semigroup (see [8]). One observes using conformal invariance that the above inequality is equivalent to the sharp Sobolev inequality on Rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization. The roots here go back to Hardy and Littlewood. The advantage of casting the problem on the sphere is that the role of the constants is evident, and one is led immediately to the conjecture that this inequality should hold whenever possible (for example, 2 < q < 0o if n = 2). This is in fact true and will be demonstrated in Section 2. A clear question at this point is "What is the situation in dimension 2?" Two important arguments ([25], [26], [27]) dealt with this issue, both motivated by geometric variational problems. Because q goes to infinity for dimension 2, the appropriate function space is the exponential class. Responding in part

The reverse isoperimetric problem for Gaussian measure

Abstract: It is shown that ifg is the standard Gaussian density on ?n andC is a convex body in ?n $$\int_{\partial C} {g \leqslant 4n^{1/4} }$$ . The arguments presented raise several questions in integral geometry.

New Gaussian estimates for enlarged balls

Abstract: Consider a centered Gaussian measure μ on a separable Banach spaceX. Denote byK the unit ball of the reproducing kernel of μ, and consider a symmetric convex setC ofX. We provide two-sided estimates of μ(C+tK). We determine in a very general setting at which rate for the gauge ofC the variablesYn(2 logn)−1/2 cluster toK, when (Yn) is an i.i.d. sequence distributed like μ. The rate depends only on the behavior of the function e→μ(eC) as e→0.

Polynomial averages in the Kontsevich model

Abstract: We obtain in closed form averages of polynomials, taken over hermitian matrices with the Gaussian measure involved in the Kontsevich integral, and prove a conjecture of Witten enabling one to express analogous averages with the full (cubic potential) measure, as derivatives of the partition function with respect to traces of inverse odd powers of the external argument. The proofs are based on elementary algebraic identities involving a new set of invariant polynomials of the linear group, closely related to the general Schur functions.

Mesures de Gauss pour des algorithmes de fractions continues multidimensionnelles

Abstract: — We give a procedure to compute invariant measures for some multi-dimensional continued fractions algorithms; we use this procedure to build in an elementary way the usual Gauss measure for the map x\->[llx], and to build an ergodic invariant measure for the algorithms of Brun and Selmer in any dimension. We show that, in the case of ordinary continued fraction, our construction admits a natural geometrie interpretation.

On Multivariate Integration for Stochastic Processes

Abstract: We present bounds on the minimal average case errors of quadrature formulas that use n function values for multivariate integration. The error bounds are derived in terms of smoothness properties of the covariance function of the underlying stochastic process.

Coherent states for the quantum complex plane

Abstract: The coherent states for the quantum complex plane are introduced. It is demonstrated that the Bargmann representation corresponding to these states involves both the standard integral with respect to the Gaussian measure and the Berezin integral over Grassmann variables. The quantum generalizations of many constructions developed for classical coherent states are described.

Ornstein-Uhlenbeck Processes as Bernstein Processes

Abstract: Bernstein diffusions belong to a new class of time symmetric (but not time homogeneous) stochastic processes associated with the quantum dynamics of nonrelativistic particles in potentials. It is shown in which sense Feynman’s formal path integral method can be interpreted in terms of those processes, specially for the subset of Gaussian Bernstein diffusions. The familiar Ornstein-Uhlenbeck process becomes, in this framework, a particular Gaussian process in a large class of Bernstein diffusions, all associated with the same dynamics. The method is also illustrated for some Hilbert space valued Gaussian Diffusions

On Radon-Nikodym derivatives of finitely-additive measures induced by nonlinear transformations on Hilbert space

Abstract: THE STUDY of finitely-additive probability measures arises in problems of modelling and estimation of stochastic signals with bounded variation paths in L,[O, T] This is because physical signals are of bounded variation and of finite energy for which L,[O, T] is the natural setting and results obtained via Ito theory hold only on a set of Wiener measure 1 It is well known that integrals of paths in &JO, T] lie in a set of Wiener measure zero Hence, there is a need to construct a theory of white noise to model large bandwidth noise arising in signal analysis with the usual properties associated with it In a series of papers [l-3], in which he advocated the use of such a framework, Balakrishnan showed that it could indeed be done and, thus, laid the basis for a such a theory He considered white noise to be the identity map on a Hilbert space H with standard Gauss measure thereon It is well known that such a measure is only finitely additive on the algebra of cylinder sets and cannot be extended to the Bore1 sets of H This line of work culminated in the excellent treatise by Kallianpur and Karandikar [4] devoted to filtering and smoothing problems The key assumption in the development of the theory in the nonlinear context so far, has been that the signal process is assumed to be defined on a countably-additive probability space with paths in H while the measurement noise process is defined on a cylindrical probability space associated with the Gauss measure The formulation does not allow for signal-noise dependence and one is forced to work with a quasi-cylindrical probability space (a product space) Despite the fact that the mathematical difficulties are daunting, there nevertheless arises the need to develop a complete theory of white noise in order to study modelling issues as well as signal-noise dependence, since these issues arise quite naturally in physical problems This paper presents one step in such a direction and is motivated by the issue of likelihood ratio evaluation for signals arising in differential systems driven by white noise The specific problem addressed in this paper is the study of Radon-Nikodym derivatives (and their evaluation) of cylindrical (or finitely-additive) measures induced by nonlinear transformations on H with standard Gauss measure thereon In the linear case this problem was solved by Balakrishnan [5] Balakrishnan [2] also obtained some results in the nonlinear case when the transformation is given by I + K where K defines a homogeneous, finite “Volterra” polynomial by exploiting the connection with the pioneering work of Cameron and Martin [6]

Integrals with Respect to Gaussian Measures

Abstract: Theory of integration w.r.t. Gaussian measure is well-developed for a wide class of linear topological spaces. This enables to investigate issues of approximate integral evaluation in this class of spaces. This chapter is devoted to construction and investigation of approximate formulae with a given algebraic degree of accuracy and of formulae in which various characteristics of Gaussian measures are used as initial data. Approximate evaluation of integrals w.r.t. Gaussian measures are considered in [40]-[58], [85]-[89].

Reidemeister torsion and simplicial quantum gravity

Abstract: We show that three-dimensional simplicial quantum gravity, as described by dynamically triangulated manifolds, is connected with a Gaussian model determined by the simple homotopy types of the underlying manifolds. By exploiting this result it is shown that the partition function of three-dimensional simplicial quantum gravity is well defined in a convex region in the plane of the gravitational and cosmological coupling constants. Such a region is determined by the Reidemeister–Franz torsion invariants associated with orthogonal representations of the fundamental groups of the set of manifolds considered. The system shows a critical behavior and undergoes a first order phase transition at a well-defined value of the couplings, again determined by the torsion invariants. On the critical line the partition function can be explicitly related to a Gaussian measure on the general linear group GL(∞, R), showing evidence of a well-defined thermodynamical limit of the theory, with a stable (vacuum) configuration corresponding to three-dimensional (homology) manifolds. The first order nature of the transition yielding such a configuration seems to support the belief in the absence of a continuum limit of the theory. More generally, the approach presented here provides further analytical support for the picture of three-dimensional simplicial quantum gravity which has been abstracted from numerical simulations.

A result on multidimensional quantization

Abstract: For any integer N > 1 , a probability space, a Gaussian random vector X defined on the space with a positive definite covariance matrix, and an N-level quantizer Q are presented such that the random vector Q(X) takes on each of the N values in its range with equal probability and such that X and Q(X) are independent.

The Poincaré inequality and entire functions

Abstract: Inequalities for spaces of entire functions on C n , which generalize the Poincare inequality for Gaussian measure, are obtained. The relationship between these inequalities and hypercontractive estimates for diffusion semi-groups are discussed

Harmonic analysis in infinite dimension

Abstract: We study the Hermite transform onL 2(μ) where μ is a Gaussian measure on a Lusin locally convex spaceE. We are then lead to a Hilbert space (ℋ) of analytic functions onE which is also a natural range for the Laplace transform. LetB be a convenient Hilbert-Schmidt operator on the Cameron-Martin spaceH of μ. There exists a natural sequence Cap n of capacities onE associated toB. This implies the Kondratev-Yokoi theorem about positive linear forms on the Hida test-functions space.

The functional integral for fields in a cavity

Abstract: The functional integral for a scalar field confined in a cavity and subjected to linear boundary conditions is discussed herein. It is shown how the functional measure can be conveniently dealt with by modifying the classical action with boundary corrections. The nonuniqueness of the boundary actions is described with a three‐parameter family of them giving identical boundary conditions. In some cases, the corresponding Green’s function will define a kind of generalized Gaussian measure on function space. The vacuum energy is discussed, paying due attention to its anomalous scale dependence, and the physical issues involved are considered.

Pathwise Nonlinear Filtering on Abstract Wiener Spaces

Abstract: The nonlinear filtering problem is studied for models where the samples of the signal and the noise are elements of some general abstract Wiener space. The signal is allowed to depend on the noise and the optimal filter is expressed as an explicit functional of the observed sample (trajectory). It is shown that this functional satisfies the Zakai equation. As a necessary technical tool, a class of shift transformations on the Wiener space is studied and an analog of Cameron-Martin-Girsanov's theorem is obtained.

The harmonic oscillator

Abstract: This chapter is the closest in these notes to what is usually called “Quantum Mechanics”. The present version is considerably shorter than the original French. It thus becomes more obvious that its main topic is not really elementary quantum mechanics, but rather elementary Fock space, and the quantum analogue of finite dimensional Gaussian random variables.

The Bellaterra connection

Abstract: A new object is introduced - the "Fischer bundle". It is, formally speaking, an Hermitean bundle of infinite rank over a bounded symmetric domain whose fibers are Hilbert spaces whose elements can be realized as entire analytic functions square integrable with respect to a Gaussian measure ("Fischer spaces"). The definition was inspired by our previous work on the "Fock bundle". An even more general framework is indicated, which allows one to look upon the two concepts from a unified point of view.

Critical Phenomena for Riemannian Manifolds: Simple Homotopy and Simplicial Quantum Gravity

Abstract: We show how Gromov's spaces of bounded geometries provide a general mathematical framework for addressing and solving many of the issues of $3D$-simplicial quantum gravity. In particular, we establish entropy estimates characterizing the asymptotic distribution of combinatorially inequivalent triangulated $3$-manifolds, as the number of tetrahedra diverges. Moreover, we offer a rather detailed presentation of how spaces of three-dimensional riemannian manifolds with natural bounds on curvatures, diameter, and volume can be used to prove that three-dimensional simplicial quantum gravity is connected to a Gaussian model determined by the simple homotopy types of the underlying manifolds. This connection is determined by a Gaussian measure defined over the general linear group $GL({\bf R},\infty)$. It is shown that the partition function of three-dimensional simplicial quantum gravity is well-defined, in the thermodynamic limit, for a suitable range of values of the gravitational and cosmological coupling constants. Such values are determined by the Reidemeister-Franz torsion invariants associated with an orthogonal representation of the fundamental groups of the set of manifolds considered. The geometrical system considered shows also critical behavior, and in such a case the partition function is exactly evaluated and shown to be equal to the Reidemeister-Franz torsion. The phase structure in the thermodynamical limit is also discussed. In particular, there are either phase transitions describing the passage from a simple homotopy type to another, and (first order) phase transitions within a given simple homotopy type which seem to confirm, on an analytical ground, the picture suggested by numerical simulations.

A nonlinear transformation of Gaussian measures

Abstract: The transformation of a measure corresponding to a Gaussian random field is considered for a special nonlinear mapping. A formula for the density of the transformed measure is given with respect to the original.