Showing papers on "Gaussian measure published in 1997"

Some Connections Between Isoperimetric and Sobolev-Type Inequalities

Abstract: Introduction Differential and integral forms of isoperimetric inequalities Proof of Theorem 1.1 A relation between the distribution of a function and its derivative A variational problem The discrete version of Theorem 5.1 Proof of propositions 1.3 and 1.5 A special case of Theorem 1.2 The uniform distribution on the sphere Existence of optimal Orlicz spaces Proof of Theorem 1.9 (the case of the sphere) Proof of Theorem 1.9 (the Gaussian case) The isoperimetric problem on the real line Isoperimetry and Sobolev-type inequalities on the real line Extensions of Sobolev-type inequalities to product measures on $\mathbf{R}^{n}$ References.

Itô-Wiener Chaos Expansion with Exact Residual and Correlation, Variance Inequalities

Abstract: We give a formula of expanding the solution of a stochastic differential equation (abbreviated as SDE) into a finite Ito-Wiener chaos with explicit residual. And then we apply this formula to obtain several inequalities for diffusions such as FKG type inequality, variance inequality and a correlation inequality for Gaussian measure. A simple proof for Houdre-Kagan's variance inequality for Gaussian measure is also given.

Fluctuations of the Phase Boundary in the 2D Ising Ferromagnet

Abstract: We discuss some statistical properties of the phase boundary in the 2D low-temperature Ising ferromagnet in a box with the two-component boundary conditions. We prove the weak convergence in C[0,1] of measures describing the fluctuations of phase boundaries in the canonical ensemble of interfaces with fixed endpoints and area enclosed below them. The limiting Gaussian measure coincides with the conditional distribution of certain Gaussian process obtained by the integral transformation of the white noise.

Limit theorems for bivariate Appell polynomials. Part I: Central limit theorems

Abstract: Consider the stationary linear process $X_t=\sum_{u=-\infty}^\infty a(t-u)\xi_u$ , $t\in {\bf Z}$ , where $\{ \xi_u\}$ is an i.i.d. finite variance sequence. The spectral density of $\{ X_t\}$ may diverge at the origin (long-range dependence) or at any other frequency. Consider now the quadratic form $Q_N=\sum_{t,s=1}^N b(t-s)P_{m,n} (X_t,X_s)$ , where $P_{m,n}(X_t,X_s)$ denotes a non-linear function (Appell polynomial). We provide general conditions on the kernels $b$ and $a$ for $N^{-1/2}Q_N$ to converge to a Gaussian distribution. We show that this convergence holds if $b$ and $a$ are not too badly behaved. However, the good behavior of one kernel may compensate for the bad behavior of the other. The conditions are formulated in the spectral domain.

Representation of a quantum field Hamiltonian inp-adic Hilbert space

Abstract: Gaussian measures on infinite-dimensional p-adic spaces are defined and the corresponding L2-spaces of p-adic-valued square integrable functions are constructed. Representations of the infinite-dimensional Weyl group are realized in such spaces and the formal analogy with the usual Segal representation is discussed. It is found that the parameters of the p-adic infinite-dimensional Weyl group are defined only on some balls. In p-adic Hilbert space, representations of quantum Hamiltonians for systems with an infinite number of degrees of freedom are constructed. The Hamiltonians with singular potentials are realized as bounded symmetric operators in L2-space with respect to a p-adic Gaussian measure.

A representation for Fermionic correlation functions

Abstract: Let dμS(a) be a Gaussian measure on the finitely generated Grassmann algebra A. Given an even W(a)∈A, we construct an operator R on A such that $$$$ for all f(a)∈A. This representation of the Schwinger functional iteratively builds up Feynman graphs by successively appending lines farther and farther from f. It allows the Pauli exclusion principle to be implemented quantitatively by a simple application of Gram's inequality.

Confidence regions for means of multivariate normal distributions and a non-symmetric correlation inequality for gaussian measure

Abstract: Let $\mu$ be a Gaussian measure (say, on ${\bf R}^n$) and let $K, L \subset {\bf R}^n$ be such that K is convex, $L$ is a "layer" (i.e. $L = \{x : a \leq \leq b \}$ for some $a$, $b \in {\bf R}$ and $u \in {\bf R}^n$) and the centers of mass (with respect to $\mu$) of $K$ and $L$ coincide. Then $\mu(K \cap L) \geq \mu(K) \cdot \mu(L)$. This is motivated by the well-known "positive correlation conjecture" for symmetric sets and a related inequality of Sidak concerning confidence regions for means of multivariate normal distributions. The proof uses an apparently hitherto unknown estimate for the (standard) Gaussian cumulative distribution function: $\Phi (x) > 1 - \frac{(8/\pi)^{{1/2}}}{3x + (x^2 +8)^{{1/2}}} e^{-x^2/2}$ (valid for $x > -1$).

Caractérisation des distributions gaussiennes sur les groupes nilpotents simplement connexes et les espaces symétriques

Abstract: For simply connected nilpotent Lie groups, we show that a probability measure is gaussian in the sense of Bernstein (for a definition thereof which in a natural way involves non-commutativity) iff it is a gaussian measure in the classical sense concentrated on an abelian subgroup. Furthermore we carry over the Skitovic-Darmois theorem to symmetric spaces of non-compact type.

RATE OF CONVERGENCE TO GAUSSIAN MEASURES ON n-SPHERES AND JACOBI HYPERGROUPS

Abstract: In this paper we prove central limit theorems of the following kind: let $S^d \subset \mathbb{R}^{d + 1}$ be the unit sphere of dimension $d \geq 2$ with uniform distribution $\omega_d$. For each $k \epsilon \mathbb{N}$, consider the isotropic random walk $(X_n^k)_{n \geq 0}$ on $S^d$ starting at the north pole with jumps of fixed sizes $\angle (X_n^k, X_{n - 1}^k) = \pi/\sqrt{k}$ for all $n \geq 1$. Then there is some $k_0(d)$ such that for all $k \geq k_0(d)$, the distributions $\varrho_k$ of $X_k^k$ have continuous, bounded $\omega_d$-densities $f_k$. Moreover, there is a (known) Gaussian measure $ u$ on $S^d$ with $\omega_d$-density such that $||f_k - h||_{\infty} = O(1/k)$ and $||\varrho_k - u|| = O(1/k)$ for $k \to \infty$, where $O(1/k)$ is sharp. We shall derive this rate of convergence in the central limit theorem more generally for a quite general class of isotropic random walks on compact symmetric spaces of rank one as well as for random walks on $[0, \pi]$ whose transition probabilities are related to product linearization formulas of Jacobi polynomials.

The change of variables formula on Wiener space

Abstract: The transformations of measure induced by a not-necessarily adapted perturbation of the identity is considered. Previous results are reviewed and recent results on absolute continuity and related Radon-Nikodym densities are derived under conditions which are ‘as near as possible’ to the conditions of Federer’s area theorem in the finite dimensional case.

On Cherednik-Macdonald-Mehta identities

Abstract: In this note we give a short proof of Cherednik's generalization of Macdonald-Mehta identities for the root system $A_{n-1}$ using the representation theory of quantum groups. These identities, suggested and proved by Cherednik, give an explicit formula for the integral of a product of Macdonald polynomials with respect to a ``difference analogue of the Gaussian measure''.

Fixed point action for the massless lattice Schwinger model

Abstract: We determine non-perturbatively the fixed-point action for fermions in the two-dimensional U(1) gauge (Schwinger) model. This is done by iterating a block spin transformation in the background of non-compact gauge field configurations sampled according to the (perfect) Gaussian measure. The resulting action has 123 independent couplings, is bilinear in the Grassmann fields, gauge-invariant by considered the compact gauge transporters and localized within a $7\times 7$ lattice centered around one of the fermions. We then simulate the model at various values of $\beta$ and compare with results obtained with the Wilson fermion action. We find excellent improvement for the studied observables (propagators and masses).