Showing papers on "Gaussian measure published in 2011"

On the isoperimetric deficit in gauss space

Abstract: We prove a sharp quantitative version of the isoperimetric inequality in the space ${\Bbb R}^n$ endowed with the Gaussian measure.

Exponential rarefaction of real curves with many components

Abstract: Given a positive real Hermitian holomorphic line bundle L over a smooth real projective manifold X, the space of real holomorphic sections of the bundle L d inherits for every d∈ℕ∗ a L 2-scalar product which induces a Gaussian measure. When X is a curve or a surface, we estimate the volume of the cone of real sections whose vanishing locus contains many real components. In particular, the volume of the cone of maximal real sections decreases exponentially as d grows to infinity.

Functional affine-isoperimetry and an inverse logarithmic Sobolev inequality

Abstract: We give a functional version of the affine isoperimetric inequality for log-concave functions which may be interpreted as an inverse form of a logarithmic Sobolev inequality inequality for entropy. A linearization of this inequality gives an inverse inequality to the Poincar'e inequality for the Gaussian measure.

Conical square functions and non-tangential maximal functions with respect to the gaussian measure

Abstract: We study, in L 1 (R n ;) with respect to the gaussian measure, nontangential maximal functions and conical square functions associated with the Ornstein-Uhlenbeck operator by developing a set of techniques which allow us, to some extent, to compensate for the non-doubling character of the gaussian measure. The main result asserts that conical square functions can be controlled in L 1 -norm by non-tangential maximal functions. Along the way we prove a change of aperture result for the latter. This complements recent results on gaussian Hardy spaces due to Mauceri and Meda.

Estimates on the condition number of random rank-deficient matrices

Abstract: Let r ≤ m ≤ n ∈ N and let A be a rank r matrix of size m x n, with entries in K = ℂ or K = R. The generalized condition number of A, which measures the sensitivity of Ker(A) to small perturbations of A, is defined as κ (A) = ∥A∥ ∥A † ∥, where † denotes Moore-Penrose pseudoinversion. In this paper we prove sharp lower and upper bounds on the probability distribution of this condition number, when the set of rank r, m x n matrices is endowed with the natural probability measure coming from the Gaussian measure in K m×n . We also prove an upper-bound estimate for the expected value of log κ in this setting.

Mass transportation and contractions

Abstract: According to a celebrated result of L. Caffarelli, every optimal mass transportation mapping pushing forward the standard Gaussian measure onto a log-concave measure $e^{-W} dx$ with $D^2 W \ge {Id}$ is 1-Lipschitz. We present a short survey of related results and various applications.

A concentration inequality for the overlap of a vector on a large set, With application to the communication complexity of the Gap-Hamming-Distance problem.

Abstract: Given two sets A; B R n , a measure of their correlation is given by the expected squared inner product between random x2 A and y2 B. We prove an inequality showing that no two sets of large enough Gaussian measure (at least e d n for some constant d > 0) can have correlation substantially lower than would two random sets of the same size. Our proof is based on a concentration inequality for the overlap of a random Gaussian vector on a large set. As an application, we show how our result can be combined with the partition bound of Jain and Klauck to give a simpler proof of a recent linear lower bound on the randomized communication complexity of the Gap-Hamming-Distance problem due to Chakrabarti and Regev.

Average Case Tractability of Non-homogeneous Tensor Product Problems

Abstract: We study d-variate approximation problems in the average case setting with respect to a zero-mean Gaussian measure. Our interest is focused on measures having a structure of non-homogeneous linear tensor product, where covariance kernel is a product of univariate kernels. We consider the normalized average error of algorithms that use finitely many evaluations of arbitrary linear functionals. The information complexity is defined as the minimal number n(h,d) of such evaluations for error in the d-variate case to be at most h. The growth of n(h,d) as a function of h^{-1} and d depends on the eigenvalues of the covariance operator and determines whether a problem is tractable or not. Four types of tractability are studied and for each of them we find the necessary and sufficient conditions in terms of the eigenvalues of univariate kernels. We illustrate our results by considering approximation problems related to the product of Korobov kernels characterized by a weights g_k and smoothnesses r_k. We assume that weights are non-increasing and smoothness parameters are non-decreasing. Furthermore they may be related, for instance g_k=g(r_k) for some non-increasing function g. In particular, we show that approximation problem is strongly polynomially tractable, i.e., n(h,d)\le C h^{-p} for all d and 0 1. For other types of tractability we also show necessary and sufficient conditions in terms of the sequences g_k and r_k.

From constructive field theory to fractional stochastic calculus. (II) Constructive proof of convergence for the L\'evy area of fractional Brownian motion with Hurst index $\alpha\in(1/8,1/4)$

Abstract: {Let $B=(B_1(t),...,B_d(t))$ be a $d$-dimensional fractional Brownian motion with Hurst index $\alpha<1/4$, or more generally a Gaussian process whose paths have the same local regularity. Defining properly iterated integrals of $B$ is a difficult task because of the low H\"older regularity index of its paths. Yet rough path theory shows it is the key to the construction of a stochastic calculus with respect to $B$, or to solving differential equations driven by $B$. We intend to show in a series of papers how to desingularize iterated integrals by a weak, singular non-Gaussian perturbation of the Gaussian measure defined by a limit in law procedure. Convergence is proved by using "standard" tools of constructive field theory, in particular cluster expansions and renormalization. These powerful tools allow optimal estimates, and call for an extension of Gaussian tools such as for instance the Malliavin calculus. After a first introductory paper \cite{MagUnt1}, this one concentrates on the details of the constructive proof of convergence for second-order iterated integrals, also known as L\'evy area.

Sobolev regularity for the Monge-Ampere equation in the Wiener space

Abstract: Given the standard Gaussian measure $\gamma$ on the countable product of lines $\mathbb{R}^{\infty}$ and a probability measure $g \cdot \gamma$ absolutely continuous with respect to $\gamma$, we consider the optimal transportation $T(x) = x + abla \varphi(x)$ of $g \cdot \gamma$ to $\gamma$. Assume that the function $| abla g|^2/g$ is $\gamma$-integrable. We prove that the function $\varphi$ is regular in a certain Sobolev-type sense and satisfies the classical change of variables formula $g = {\det}_2(I + D^2 \varphi) \exp \bigl(\mathcal{L} \varphi - 1/2 | abla \varphi|^2 \bigr)$. We also establish sufficient conditions for the existence of third order derivatives of $\varphi$.

Gradient Flow from a Random Walk in Hilbert Space

Abstract: Consider a probability measure on a Hilbert space defined via its density with respect to a Gaussian. The purpose of this paper is to demonstrate that an appropriately defined Markov chain, which is reversible with respect to the measure in question, exhibits a diffusion limit to a noisy gradient flow, also reversible with respect to the same measure. The Markov chain is defined by applying a Metropolis-Hastings accept-reject mechanism to an Ornstein-Uhlenbeck proposal which is itself reversible with respect to the underlying Gaussian measure. The resulting noisy gradient flow is a stochastic partial differential equation driven by a Wiener process with spatial correlation given by the underlying Gaussian structure.

Generalised particle filters with Gaussian measures

Abstract: The stochastic filtering problem deals with the estimation of the posterior distribution of the current state of a signal process X = {X t } t≥0 given the information supplied by an associate process Y ={Y t } t≥0 . The scope and range of its applications includes the control of engineering systems, global data assimilation in meteorology, volatility estimation in financial markets, computer vision and vehicle tracking. A massive scientific and computational effort is dedicated to the development of viable tools for approximating the solution of the filtering problem. Classical PDE methods can be successful, particularly if the state space has low dimensions. In higher dimensions, a class of numerical methods called particle filters have proved the most successful methods to-date. These methods produce an approximations of the posterior distribution by using the empirical distribution of a cloud of particles that explore the signal's state space. We discuss here a more general class of numerical methods which involve generalised particles, that is, particles that evolve through larger spaces. Such generalised particles include Gaussian measures, wavelets, and finite elements in addition to the classical particle methods. We will construct the approximating particle system under the Gaussian measure framework and prove the corresponding convergence result.

Laplace-type exact asymptotic formulas for the Bogoliubov Gaussian measure

Abstract: We obtain new asymptotic formulas for two classes of Laplace-type functional integrals with the Bogoliubov measure. The principal functionals are the Lp functionals with 0 < p < ∞ and two functionals of the exact-upper-bound type. In particular, we prove theorems on the Laplace-type asymptotic behavior for the moments of the Lp norm of the Bogoliubov Gaussian process when the moment order becomes infinitely large. We establish the existence of the threshold value p 0 = 2+4π 2 /β 2 ω 2 , where β > 0 is the inverse temperature and ω > 0 is the harmonic oscillator eigenfrequency. We prove that the asymptotic behavior under investigation differs for 0 < p < p 0 and p > p 0 . We obtain similar asymptotic results for large deviations for the Bogoliubov measure. We establish the scaling property of the Bogoliubov process, which allows reducing the number of independent parameters.

From Constructive Field Theory to Fractional Stochastic Calculus. (I) An introduction: Rough Path Theory and Perturbative Heuristics

Abstract: Let B = (B 1(t), . . . , B d (t)) be a d-dimensional fractional Brownian motion with Hurst index α ≤ 1/4, or more generally a Gaussian process whose paths have the same local regularity. Defining properly iterated integrals of B is a difficult task because of the low Holder regularity index of its paths. Yet rough path theory shows it is the key to the construction of a stochastic calculus with respect to B, or to solving differential equations driven by B. We intend to show in a forthcoming series of papers how to desingularize iterated integrals by a weak singular non-Gaussian perturbation of the Gaussian measure defined by a limit in law procedure. Convergence is proved by using “standard” tools of constructive field theory, in particular cluster expansions and renormalization. These powerful tools allow optimal estimates of the moments and call for an extension of the Gaussian tools such as for instance the Malliavin calculus. This first paper aims to be both a presentation of the basics of rough path theory to physicists, and of perturbative field theory to probabilists; it is only heuristic, in particular because the desingularization of iterated integrals is really a non-perturbative effect. It is also meant to be a general motivating introduction to the subject, with some insights into quantum field theory and stochastic calculus. The interested reader should read for a second time the companion article (Magnen and Unterberger in From constructive theory to fractional stochastic calculus. (II) The rough path for $${\frac{1}{6} < \alpha < \frac{1}{4}}$$ : constructive proof of convergence, 2011, preprint) for the constructive proofs.

Mixing operators and small subsets of the circle

Abstract: We provide complete characterizations, on Banach spaces with cotype 2, of those linear operators which happen to be weakly mixing or strongly mixing transformations with respect to some nondegenerate Gaussian measure. These characterizations involve two families of small subsets of the circle: the countable sets, and the so-called sets of uniqueness for Fourier-Stieltjes series. The most interesting part, i.e. the sufficient conditions for weak and strong mixing, is valid on an arbitrary (complex, separable) Frechet space.

Gaussian measures of dilations of convex rotationally symmetric sets in $\mathbb{C}^n$

Abstract: We consider the complex case of the S-inequality . It concerns the behaviour of Gaussian measures of dilations of convex and rotationally symmetric sets in $\mathbb{C}^n$. We pose and discuss a conjecture that among all such sets measures of cylinders decrease the fastest under dilations. Our main result in this paper is that this conjecture holds under the additional assumption that the Gaussian measure of the sets considered is not greater than some constant $c > 0.64$.

On the random walk metropolis algorithm for Gaussian random field priors and the gradient flow

Abstract: Consider a probability measure on a Hilbert space defined via its density with respect to a Gaussian. The purpose of this paper is to demonstrate that an appropriately defined Markov chain, which is reversible with respect to the measure in question, exhibits a diffusion limit to a noisy gradient flow, also reversible with respect to the same measure. The Markov chain is defined by applying a Metropolis-Hastings accept-reject mechanism to an Ornstein-Uhlenbeck proposal which is itself reversible with respect to the underlying Gaussian measure. The resulting noisy gradient flow is a stochastic partial differential equation driven by a Wiener process with spatial correlation given by the underlying Gaussian structure.

Quantitative Isoperimetric Inequalities on the Real Line

Abstract: In a recent paper A. Cianchi, N. Fusco, F. Maggi, and A. Pratelli have shown that, in the Gauss space, a set of given measure and almost minimal Gauss boundary measure is necessarily close to be a half-space. Using only geometric tools, we extend their result to all symmetric log-concave measures \mu on the real line. We give sharp quantitative isoperimetric inequalities and prove that among sets of given measure and given asymmetry (distance to half line, i.e. distance to sets of minimal perimeter), the intervals or complements of intervals have minimal perimeter.

The Gaussian Free Field

Abstract: By a well known calculation on Gaussian measure, if X is finite, for any \({X}{\varepsilon}{\mathbb{R}}^{X}_{+}\)\(\frac{\sqrt{\det(M_{\lambda}-C)}}{(2\pi)^{\left| X\right| /2}}\int_{\mathbb{R}^{X}}e^{-\frac{1}{2}\sum\chi_{u}(v^{u})^{2}}e^{-\frac{1} {2}e(v)}\Pi_{u\in X}dv^{u}=\sqrt{\frac{\det(G_{\chi})}{\det(G)}}\) and \(\frac{\sqrt{\det(M_{\lambda}-C)}}{(2\pi)^{\left| X\right| /2}}\int_{\mathbb{R}^{X}}v^xv^ye^{-\frac{1}{2}\sum\chi_{u}(v^{u})^{2}}e^{-\frac{1} {2}e(v)}\Pi_{u\in X}dv^{u}=(G_{\chi})^{x,y}\sqrt{\frac{\det(G_{\chi})}{\det(G)}} \)

Optimal Proposal Design for Random Walk Type Metropolis Algorithms with Gaussian Random Field Priors

Abstract: Consider a probability measure on a Hilbert space defined via its density with respect to a Gaussian. The purpose of this paper is to demonstrate that an appropriately defined Markov chain, which is reversible with respect to the measure in question, exhibits a diffusion limit to a noisy gradient flow, also reversible with respect to the same measure. The Markov chain is defined by applying a Metropolis-Hastings accept-reject mechanism to an Ornstein-Uhlenbeck proposal which is itself reversible with respect to the underlying Gaussian measure. The resulting noisy gradient flow is a stochastic partial differential equation driven by a Wiener process with spatial correlation given by the underlying Gaussian structure.

Solution to the Navier-Stokes equations with random initial data

Abstract: We construct a solution to the spatially periodic $d$-dimensional Navier-Stokes equations with a given distribution of the initial data. The solution takes values in the Sobolev space $H^\alpha$, where the index $\alpha\in R$ is fixed arbitrary. The distribution of the initial value is a Gaussian measure on $H^\alpha$ whose parameters depend on $\alpha$. The Navier-Stokes solution is then a stochastic process verifying the Navier-Stokes equations almost surely. It is obtained as a limit in distribution of solutions to finite-dimensional ODEs which are Galerkin-type approximations for the Navier-Stokes equations. Moreover, the constructed Navier-Stokes solution $U(t,\omega)$ possesses the property: $$E[f(U(t,\omega))] = \int_{H^\alpha} f(e^{t u\Delta} u) \gamma(du)$$, where $f \in L_1(\gamma)$, $e^{t \Delta}$ is the heat semigroup, $ u$ is the viscosity in the Navier-Stokes equations, and $\gamma$ is the distribution of the initial data.

On short-time asymptotics of one-dimensional Harris flows

Abstract: We study the short-time asymptotical behavior of stochastic flows onRin the sup-norm. The results are stated in terms of a Gaussian processassociated with the covariation of the flow. In case the Gaussian processhas a continuous version the two processes can be coupled in such a waythat the difference is uniformly o√tlnlnt −1 . In case it has no continu-ous version, an O√tlnlnt −1 estimate is obtained under mild regularityassumptions. The main tools are Gaussian measure concentration and amartingale version of the Slepian comparison principle.Keywords: stochastic flows, law of iterated logarithm, Slepian com-parison2010 AMS Math subject classification: 60G17, 60G44 1 Introduction In this paper we investigate the asymptotical behaviour of the point motion ofone-dimensional stochastic flows. The term “stochastic flow” means a family ofrandom maps (X s,t (·)) s≤t that satisfies the flow property X t,r ◦X s,t = X s,r andhas independent values on disjoint intervals. What we call the point motion isthe family of maps X

The Monge problem in Wiener Space

Abstract: We address the Monge problem in the abstract Wiener space and we give an existence result provided both marginal measures are absolutely continuous with respect to the infinite dimensional Gaussian measure {\gamma}.

Clark-Ocone type formulas in the Meixner white noise analysis

Abstract: In the classical Gaussian analysis the Clark-Ocone formula allows to reconstruct an integrand if we know the Ito stochastic integral. This formula can be written in the form $$ F=\mathbf EF+\int\mathbf E\big\{\partial_t F|_{\mathcal F_t}\big\} dW_t, $$ where a function (a random variable) $F$ is square integrable with respect to the Gaussian measure and differentiable by Hida; $\mathbf E$ $-$ the expectation; $\mathbf E\big\{\circ|_{\mathcal F_t}\big\}$ $-$ the conditional expectation with respect to a full $\sigma$-algebra $\mathcal F_t$ that is generated by the Wiener process $W$ up to the point of time $t$; $\partial_\cdot F$ $-$ the Hida derivative of $F$; $\int\circ (t)dW_t$ $-$ the Ito stochastic integral with respect to the Wiener process. In this paper, we explain how to reconstruct an integrand in the case when instead of the Gaussian measure one considers the so-called generalized Meixner measure $\mu$ (depending on parameters, $\mu$ can be the Gaussian, Poissonian, Gamma measure etc.) and obtain corresponding Clark-Ocone type formulas.

Representation of Conditional Expectations in Gaussian Analysis on Sequence Spaces

Abstract: From a given nuclear triplet we construct a nuclear triplet of sequence spaces and introduce a correlated Gaussian measure via the Bochner-Minlos theorem. Considering special types of correlation operators on such sequence spaces, certain conditional expectations can be given in an explicit way.

Bounds for maximal functions associated to rotational invariant measures in high dimensions

Abstract: In recent articles it was proved that when $\mu$ is a finite, radial measure in $\real^n$ with a bounded, radially decreasing density, the $L^p(\mu)$ norm of the associated maximal operator $M_\mu$ grows to infinity with the dimension for a small range of values of $p$ near 1. We prove that when $\mu$ is Lebesgue measure restricted to the unit ball and $p 2$, when restricted to radially decreasing functions. On the other hand, when $\mu$ is the Gaussian measure, the $L^p$ operator norms of the maximal operator grow to infinity with the dimension for any finite $p> 1$, even in the subspace of radially decreasing functions.