Search

This paper is concerned with an inverse random source problem for the one-dimensional stochastic Helmholtz equation with attenuation. The source is assumed to be a microlocally isotropic Gaussian random field with its covariance operator being a classical pseudo-differential operator. The random sources under consideration are equivalent to the generalized fractional Gaussian random fields which include rough fields and can be even rougher than the white noise, and hence should be interpreted as distributions. The well-posedness of the direct source problem is established in the distribution sense. The micro-correlation strength of the random source, which appears to be the strength in the principal symbol of the covariance operator, is proved to be uniquely determined by the wave field in an open measurement set. Numerical experiments are presented for the white noise model to demonstrate the validity and effectiveness of the proposed method.

In this paper we will be concerned with the problem of the existence of an invariant mixing measure considering its connection with the chaotic behavior of linear semigroups on separable Banach spaces. We first prove an identity characterizing invariant Gaussian measure involving its covariance operator and the infinitesimal generator of the semigroup. This gives an answer to a question raised by Rudnicki in his inspiring review paper [32] . Under suitable conditions, we use the proved identity to give an invariant mixing Gaussian measure as distribution of a Wiener integral.

This paper is concerned with the finite element approximation of the stochastic Cahn–Hilliard–Cook equation driven by an infinite dimensional Wiener type noise. The Argyris finite elements are used to discretize the spatial variables while the infinite dimensional (cylindrical) Wiener process is approximated by truncated stochastic series spanned by the spectral basis of the covariance operator. The optimal strong convergence order in L 2 and H ˙ − 2 norms is obtained. Unlike the mixed finite element method studied in the existing literature, our method allows the covariance operator of the Wiener process to have an infinite trace, including the space–time white noise is allowed in our model. Numerical experiments are presented to illustrate the theoretical analysis.

In this article, we introduce the functional envelope for sufficient dimension reduction and regression with functional and longitudinal data. Functional sufficient dimension reduction methods, especially the inverse regression estimation family of methods, usually involve solving generalized eigenvalue problems and inverting the infinite-dimensional covariance operator. With the notion of functional envelope, essentially a special type of sufficient dimension reduction subspace, we develop a generic method to circumvent the difficulties in solving the generalized eigenvalue problems and inverting the covariance directly. We derive the geometric characteristics of the functional envelope and establish the asymptotic properties of related functional envelope estimators under mild conditions. The functional envelope estimators have shown promising performance in extensive simulation studies and real data analysis.

This paper addresses the exponential stability of the trivial solution of some types of evolution equations driven by Holder continuous functions with Holder index greater than 1/2. The results can be applied to the case of equations whose noisy inputs are given by a fractional Brownian motion B H with covariance operator Q, provided that H ∈ ( 1 / 2 , 1 ) and tr ( Q ) is sufficiently small.

In this paper, we establish the sharp L p boundedness for the bilinear integral operators with certain hypersingularities that generalize the bilinear Hilbert transform.

We extend the notion of cointegration for multivariate time series to a potentially infinite-dimensional setting in which our time series takes values in a complex separable Hilbert space. In this setting, standard linear processes with nonzero long-run covariance operator play the role of I0 processes. We show that the cointegrating space for an I1 process may be sensibly defined as the kernel of the long-run covariance operator of its difference. The inner product of an I1 process with an element of its cointegrating space is a stationary complex-valued process. Our main result is a version of the Granger–Johansen representation theorem: we obtain a geometric reformulation of the Johansen I(1) condition that extends naturally to a Hilbert space setting, and show that an autoregressive Hilbertian process satisfying this condition, and possibly also a compactness condition, admits an I1 representation.

Unlike classical principal component analysis (PCA) for multivariate data, one needs to smooth or regularize when estimating functional principal components. Silverman's method for smoothed functional principal components has nice theoretical and practical properties. Some theoretical properties of Silverman's method were obtained using tools in the L 2 and the Sobolev spaces. This paper proposes an approach, in a general manner, to study the asymptotic properties of Silverman's method in an abstract Hilbert space. This is achieved by exploiting the perturbation results of the eigenvalues and the corresponding eigenvectors of a covariance operator. Consistency and asymptotic distributions of the estimators are derived under mild conditions. First we restrict our attention to the first smoothed functional principal component and then extend the same method for the first K smoothed functional principal components.

In this article, we investigate the asymptotic behavior of the solution to a one-dimensional stochastic heat equation with random nonlinear term generated by a stationary, ergodic random field. We extend the well-known central limit theorem for finite-dimensional diffusions in random environment to this infinite-dimensional setting. Due to our result, a central limit theorem in $L^1$ sense with respect to the randomness of the environment holds under a diffusive time scaling. The limit distribution is a centered Gaussian law whose covariance operator is explicitly described. It concentrates only on the space of constant functions., 20 pages

We study an inverse acoustic scattering problem in half-space with a probabilistic impedance boundary value condition. The Robin coefficient (surface impedance) is assumed to be a Gaussian random function $\lambda = \lambda(x)$ with a pseudodifferential operator describing the covariance. We measure the amplitude of the backscattered field averaged over the frequency band and assume that the data is generated by a single realization of $\lambda$. Our main result is to show that under certain conditions the principal symbol of the covariance operator of $\lambda$ is uniquely determined. Most importantly, no approximations are needed and we can solve the full non-linear inverse problem. We concentrate on anisotropic models for the principal symbol, which leads to the analysis of a novel anisotropic spherical Radon transform and its invertibility.

A basic problem in operator theory is to estimate how a small perturbation effects the eigenspaces of a self-adjoint compact operator. In this paper, we prove upper bounds for the subspace distance, taylored for structured random perturbations. As a main example, we consider the empirical covariance operator, and show that a sharp bound can be achieved under a relative gap condition. The proof is based on a novel contraction phenomenon, contrasting previous spectral perturbation approaches., Comment: 15 pages

This paper is concerned with transition paths within the framework of the overdamped Langevin dynamics model of chemical reactions. We aim to give an efficient description of typical transition paths in the small temperature regime. We adopt a variational point of view and seek the best Gaussian approximation, with respect to Kullback-Leibler divergence, of the non-Gaussian distribution of the diffusion process. We interpret the mean of this Gaussian approximation as the "most likely path" and the covariance operator as a means to capture the typical fluctuations around this most likely path. We give an explicit expression for the Kullback-Leibler divergence in terms of the mean and the covariance operator for a natural class of Gaussian approximations and show the existence of minimisers for the variational problem. Then the low temperature limit is studied via $\Gamma$-convergence of the associated variational problem. The limiting functional consists of two parts: The first part only depends on the mean and coincides with the $\Gamma$-limit of the Freidlin-Wentzell rate functional. The second part depends on both, the mean and the covariance operator and is minimized if the dynamics are given by a time-inhomogenous Ornstein-Uhlenbeck process found by linearization of the Langevin dynamics around the Freidlin-Wentzell minimizer., Comment: 42 pages

Infinite dimensional periodically correlated (PC) random fields are studied in spectral domain. A spectral characterization is given and harmonizability is established. The covariance operator is characterized where it is observed that an infinite dimensional PC field is a two-dimensional Fourier transform of a spectral random measure. Also, an evolutionary spectral representation and a space-dependent spectral density are given.

International audience; We prove global well-posedness of the subcritical generalized Korteweg-de Vries equation (the mKdV and the gKdV with quartic power of nonlinearity) subject to an additive random perturbation. More precisely, we prove that if the driving noise is a cylindrical Wiener process on L 2 (R) and the covariance operator is Hilbert-Schmidt in an appropriate Sobolev space, then the solutions with H 1 (R) data are globally wellposed in H 1 (R). This extends results obtained by A. de Bouard and A. Debussche for the stochastic KdV equation. Dedication: In the memory of Igor Chueshov.

The aim of this paper is to investigate the pathwise numerical solution of semilinear parabolic stochastic partial differential equations (SPDEs) with colored noise instead of the usual space–time white noise. We estimate the numerical solution in the L∞ topology by a method that takes advantages of the smoothing effect of the dominant linear operator. We consider the case the covariance operator of the forcing does not necessarily commute with the linear operator of the SPDE because of the fact that the Brownian motions are not necessarily independent. We show convergence of this method, and numerical examples give insight into the reliability of the theoretical study. Copyright © 2015 John Wiley & Sons, Ltd.

We obtain new upper tail probabilities of $m$-times integrated Brownian motions under the uniform norm and the $L^p$ norm. For the uniform norm, Talagrand's approach is used, while for the $L^p$ norm, Zolotare's approach together with suitable metric entropy and the associated small ball probabilities are used. This proposed method leads to an interesting and concrete connection between small ball probabilities and upper tail probabilities (large ball probabilities) for general Gaussian random variable in Banach spaces. As applications, explicit bounds are given for the largest eigenvalue of the covariance operator, and appropriate limiting behaviors of the Laplace transforms of $m$-times integrated Brownian motions are presented as well.

This paper deals with analyzing structural breaks in the covariance operator of sequentially observed functional data. For this purpose, procedures are developed to segment an observed stretch of curves into periods for which second-order stationarity may be reasonably assumed. The proposed methods are based on measuring the fluctuations of sample eigenvalues, either individually or jointly, and traces of the sample covariance operator computed from segments of the data. To implement the tests, new limit results are introduced that deal with the large-sample behavior of vector-valued processes built from partial sample eigenvalue estimates. These results in turn enable the calibration of the tests to a prescribed asymptotic level. A simulation study and an application to Australian annual minimum temperature curves confirm that the proposed methods work well in finite samples. The application suggests that the variation in annual minimum temperature underwent a structural break in the 1950s, after which typical fluctuations from the generally increasing trendstarted to be significantly smaller., Comment: 33 Pages

We study a problem of estimation of smooth functionals of parameter $\theta $ of Gaussian shift model $$ X=\theta +\xi,\ \theta \in E, $$ where $E$ is a separable Banach space and $X$ is an observation of unknown vector $\theta$ in Gaussian noise $\xi$ with zero mean and known covariance operator $\Sigma.$ In particular, we develop estimators $T(X)$ of $f(\theta)$ for functionals $f:E\mapsto {\mathbb R}$ of H\"older smoothness $s>0$ such that $$ \sup_{\|\theta\|\leq 1} {\mathbb E}_{\theta}(T(X)-f(\theta))^2 \lesssim \Bigl(\|\Sigma\| \vee ({\mathbb E}\|\xi\|^2)^s\Bigr)\wedge 1, $$ where $\|\Sigma\|$ is the operator norm of $\Sigma,$ and show that this mean squared error rate is minimax optimal at least in the case of standard Gaussian shift model ($E={\mathbb R}^d$ equipped with the canonical Euclidean norm, $\xi =\sigma Z,$ $Z\sim {\mathcal N}(0;I_d)$). Moreover, we determine a sharp threshold on the smoothness $s$ of functional $f$ such that, for all $s$ above the threshold, $f(\theta)$ can be estimated efficiently with a mean squared error rate of the order $\|\Sigma\|$ in a "small noise" setting (that is, when ${\mathbb E}\|\xi\|^2$ is small). The construction of efficient estimators is crucially based on a "bootstrap chain" method of bias reduction. The results could be applied to a variety of special high-dimensional and infinite-dimensional Gaussian models (for vector, matrix and functional data)., Comment: 49 pages

In this article, we give some characterizations of Lipschitz space via commutators of bilinear singular integral operators and bilinear fractional integral operators, respectively.

In this thesis, three open problems related to the isotropic positive definite functions on the sphere have been solved. Firstly, we provide necessary and sufficient conditions for the equivalence of two Gaussian measures with two different covariance models with associated d-Schoenberg sequences. Secondly, we find the d-Schoenberg coefficients associated with an isotropic positive definite function on the sphere in terms of the Fourier coefficients on the circle. Thirdly, we drive the infimum of the second derivative at zero for the class of the locally supported isotropic positive definite functions on the sphere. Finally, we propose a new family of Markov processes in maximal compact subgroups of a semisimple Lie group. Programa de Doctorat en Ciències

We survey some aspects of the classical prediction theory for stationary processes, in discrete time in Section 1, turning in Section 2 to continuous time, with particular reference to reproducing-kernel Hilbert spaces and the sampling theorem. We discuss the discrete-continuous theories of ARMA-CARMA, GARCH-COGARCH, and OPUC-COPUC in Section 3. We compare the various models treated in Section 4 by how well they model volatility, in particular volatility clustering. We discuss the infinite-dimensional case in Section 5, and turn briefly to applications in Section 6.

We investigate the bilinear Hilbert transform with oscillatory factors and the truncated bilinear Hilbert transform. The main result is that theLp1×Lp2→Lq-boundedness of the two operators is equivalent with1≤p1,p2<∞, and1/q=1/p1+1/p2. In addition, we also discuss the boundedness of a variant operator of bilinear Hilbert transform with a nontrivial polynomial phase.

We present and analyze an infinite dimensional Bayesian inference formulation, and its numerical approximation, for the inverse problem of inferring the shape of an obstacle from scattered acoustic waves. Given a Gaussian prior measure on the shape space, whose covariance operator is the inverse of the Laplacian, the Bayesian solution of the inverse problem is the posterior measure given by the Radon-Nikodym derivative with respect to the Gaussian prior measure. The well-posedness of the Bayesian formulation in infinite dimensions is proved, including the justification of the Radon- Nikodym derivative and the continuous dependence of the posterior measure on the observation data via the Hellinger distance. The proof is made possible by a suitable shape parametrization in a Banach space setting and the regularity of the forward solution with respect to the smoothness of the shape. This also facilitates proving the Lipschitz continuity of the observation operator with respect to the scatterer shape via the shape derivative of the forward solution. Next, a finite dimensional approximation to the Bayesian posterior is proposed and the corresponding approximation error is quantified. The approximation strategy involves a Nystrom scheme for approximating a boundary integral formulation of the forward Helmholtz problem and a Karhunen-Loeve truncation for ap- proximating the prior measure. Weak convergence of the resulting finite dimensional approximation, as well as convergence in the Hellinger distance, are investigated. In particular, we determine the convergence rate as a function of the number of Nystrom quadrature points and the number of trun- cated terms in the Karhunen-Loeve series. Finally, we estimate the error between the exact posterior moments, e.g., posterior mean and variance, and their finite dimensional approximate counterparts in terms of the errors due to forward equation approximation and prior approximation. The main result of this work is that the convergence rate for approximating the Bayesian inverse problem is spectral, and this directly inherits the spectral convergence rates of the approximations of both the prior and the forward problems.

Space-time variational formulations and adaptive Wiener–Hermite polynomial chaos Galerkin discretizations of Kolmogorov equations in infinite dimensions, such as Fokker–Planck and Ornstein–Uhlenbeck equations for functions defined on an infinite-dimensional separable Hilbert space \(H\), are developed. The well-posedness of these equations in the Hilbert space \(\mathrm{L}^2(H,\mu )\) of functions on the infinite-dimensional domain \(H\), which are square-integrable with respect to a Gaussian measure \(\mu \) with trace class covariance operator \(Q\) on \(H\), is proved. Specifically, for the infinite-dimensional Fokker–Planck equation, adaptive space-time Galerkin discretizations, based on a wavelet polynomial chaos Riesz basis obtained by tensorization of biorthogonal piecewise polynomial wavelet bases in time with a spatial Wiener–Hermite polynomial chaos arising from the Wiener–Ito decomposition of \(\mathrm{L}^2(H,\mu )\), are introduced. The resulting space-time adaptive Wiener–Hermite polynomial Galerkin discretization algorithms of the infinite-dimensional PDE are proved to converge quasioptimally in the sense that they produce sequences of finite-dimensional approximations that attain the best possible convergence rates afforded by best \(N\)-term approximations of the solution from tensor-products of multiresolution (wavelet) time-discretizations and the Wiener–Hermite polynomial chaos in \(\mathrm{L}^2(H,\mu )\). As a consequence, the proposed adaptive Galerkin solution algorithms exhibit dimension-independent performance, which is optimal with respect to the algebraic best \(N\)-term rate afforded by the solution and the polynomial degree and regularity of the multiresolution (wavelet) time-discretizations in the finite-dimensional case, in particular. All constants in our error and complexity bounds are shown to be independent of the number of “active” coordinates identified by the proposed adaptive Galerkin approximation algorithms. The computational work and memory required by the proposed algorithms scale linearly with the support size of the coefficient vectors that arise in the approximations, with dimension-independent constants.

Kernel canonical correlation analysis (CCA) is a nonlinear extension of CCA. It is widely used in information retrieval. However, relatively little research concerning the convergence rate and the distance between two feature spaces has been done so far. This paper gives the distance measure between two subspaces which was spanned by the eigenfunctions corresponding to the m largest eigenvalues of normalized cross-covariance operator (NOCCO) and its empirical version (empirical NOCCO) respectively. We established that the minimal distance between the above two spaces depends on two parameters, one is the decay rate of regularization parameter, the other is the decay rate of NOCCO compared with the eigenvalue and eigenfunctions of covariance operators.

This is a study of principal component analysis performed on a statistical sample. We assume that this data sample is made of independent copies of some random variable ranging in a separable real Hilbert space. This covers data in function spaces as well as data represented in reproducing kernel Hilbert spaces. Based on some new inequalities about the perturbation of nonnegative self-adjoint operators, we provide new bounds for the statistical fluctuations of the principal component representation with the draw of the statistical sample. ¶ We suggest two kinds of improvements to decrease these fluctuations: the first is to use a robust estimate of the covariance operator, for which non-asymptotic bounds of the estimation error are available under weak polynomial moment assumptions. The second improvement is to use some modification of the projection on the principal components based on functional calculus applied to the covariance operator. Using this modified projection, we can obtain bounds that do not depend on the spectral gap but on some more favorable factor. ¶ In appendix, we provide a new approach to the analysis of the relative positions of two orthogonal projections that is useful for our proofs and that has an interest of its own.

This work reviews recent results on an asymptotic factorization of the Small–Ball Probability of a \( {\fancyscript{L}}_{[0,1]}^{2} \)–valued process, as the radius of the ball tends to zero. This factorization involves a volumetric term, a pseudo–density for the probability law of the process, and a correction factor. Estimators of the latter two factors are introduced and some of their theoretical properties considered.

We derive estimates for the rate of strong Gaussian approximation in the invariance principle in a Hilbert space for sums of i.i.d. random vectors. It is shown that the estimates are optimal with respect to other if the sequence of eigenvalues of the covariance operator of summands decreases slowly.

We find exact small deviation asymptotics with respect to a weighted Hilbert norm for some well-known Gaussian processes. Our approach does not require knowledge of the eigenfunctions of the covariance operator of a weighted process. Such a peculiarity of the method makes it possible to generalize many previous results in this area. We also obtain new relations connected to exact small deviation asymptotics for a Brownian excursion, a Brownian meander, and Bessel processes and bridges.

We show how to integrate operator valued functions with respect to a spectral or orthogonally scattered measure. Such measures typically have a variation which has either the value zero or infinity on any set and cannot therefore be treated by the approaches of Bartle or Dobrakov. Bilinear integrals of this type arise from trace class operators between Banach function spaces and in the connection between stationary-state scattering theory and time-dependent scattering theory in Hilbert space.

We observe two sequences of curves which are connected via an integral operator. Our model includes linear models as well as autoregressive models in Hilbert spaces. We wish to test the null hypothesis that the operator did not change during the observation period. Our method is based on projecting the observations onto a suitably chosen finite dimensional space. The testing procedure is based on functionals of the weighted residuals of the projections. Since the quadratic form is based on estimating the long-term covariance matrix of the residuals, we also provide some results on Bartlett-type estimators.

Stability of a nonlinear elastic beam excited by a space-time dependent nonrandom force or a L 2 -regular random noise f ( x , t ) (white in time, with general spatial covariance) is considered. It is supposed that the transversal displacement y of the beam with length l is governed by the quasilinear partial differential equation (PDE) ρ 0 h y t t + D y x x x x − [ N 0 + e ∫ 0 l ( y x ′ ) 2 d x ′ ] y x x + k y = f ( x , t ) , 0 x l , t ≥ 0 . As a main result we show that its expected energy is linearly bounded at time t in spite of the presence of additive L 2 -regular space-time noise and cubic-type nonlinearities. Appropriate partial-implicit discretization with similar qualitative behavior, consistency and stability are discussed as well. The technique of Lyapunov-type functionals processing the information on the related energy E is exploited. We compare our results to those for the linear beam. In both cases, the expected energy E [ E ] ( t ) is nondecreasing and growing linearly in time t . In fact, for all nonrandom t ≥ s and additive L 2 -regular noise f with covariance operator Q ( f ) with finite trace, it satisfies the trace formula E [ E ] ( t ) = E [ E ] ( s ) + Trace ( Q ( f ) ) ( t − s ) / 2 .

The linearized Cahn–Hilliard–Cook equation is discretized in the spatial variables by a standard finite-element method. Strong convergence estimates are proved under suitable assumptions on the covariance operator of the Wiener process, which is driving the equation. Backward Euler time stepping is also studied. The analysis is set in a framework based on analytic semigroups. The main effort is spent on proving detailed error bounds for the corresponding deterministic Cahn–Hilliard equation. The results should be interpreted as results on the approximation of the stochastic convolution, which is a part of the mild solution of the nonlinear Cahn–Hilliard–Cook equation.

We consider periodically correlated autoregressive processes in Hilbert spaces. Our studies on these processes involve existence, covariance structure, estimation of the covariance operators, strong law of large numbers and central limit theorem.

We introduce the covariance of a number of given shapes if they are interpreted as boundary contours of elastic objects. Based on the notion of nonlinear elastic deformations from one shape to another, a suitable linearization of geometric shape variations is introduced. Once such a linearization is available, a principal component analysis can be investigated. This requires the definition of a covariance metric--an inner product on linearized shape variations. The resulting covariance operator robustly captures strongly nonlinear geometric variations in a physically meaningful way and allows to extract the dominant modes of shape variation. The underlying elasticity concept represents an alternative to Riemannian shape statistics. In this paper we compare a standard L 2-type covariance metric with a metric based on the Hessian of the nonlinear elastic energy. Furthermore, we explore the dependence of the principal component analysis on the type of the underlying nonlinear elasticity. For the built-in pairwise elastic registration, a relaxed model formulation is employed which allows for a non-exact matching. Shape contours are approximated by single well phase fields, which enables an extension of the method to a covariance analysis of image morphologies. The model is implemented with multilinear finite elements embedded in a multi-scale approach. The characteristics of the approach are demonstrated on a number of illustrative and real world examples in 2D and 3D.

Pseudodifferential evolution models have been widely used in the description of biological, geophysical and environmental systems. We consider the case where functional sample information is available from such systems. Specifically, the observation model is defined in terms of a sequence of spatial realizations of the process of interest, solution to a spatiotemporal pseudodifferential equation, affected by additive strong Hilbertian white noise. In this paper, conditions for a stable computation of the solution to the associated functional filtering problem are established in terms of the covariance operator spectra of the process of interest and of the Hilbertian observation noise. In practice, such conditions are referred to the empirical spectra associated with the covariance operator estimators. A simulation study is developed to illustrate the results derived regarding robustness of the functional estimator against functional variability of the data.

We establish some estimates on the hyper bilinear Hilbert transform on both Euclidean space and torus. We also use a transference method to obtain a Kenig-Stein's estimate on bilinear fractional integrals on the n-torus.

We consider a semilinear parabolic PDE driven by additive noise. The equation is discretized in space by a standard piecewise linear finite element method. We show that the orthogonal expansion of the finite-dimensional Wiener process, that appears in the discretized problem, can be truncated severely without losing the asymptotic order of the method, provided that the kernel of the covariance operator of the Wiener process is smooth enough. For example, if the covariance operator is given by the Gauss kernel, then the number of terms to be kept is the quasi-logarithm of the number of terms in the original expansion. Then one can reduce the size of the corresponding linear algebra problem enormously and hence reduce the computational complexity, which is a key issue when stochastic problems are simulated.

We give one sufficient and two necessary conditions for boundedness between Lebesgue or Lorentz spaces of several classes of bilinear multiplier operators closely connected with the bilinear Hilbert transform.

In this note, we shall study a class of retarded Langevin equations in infinite-dimensional spaces. An informative description of stationary solutions is given by means of Green operators, Lyapunov equations, the second moment bounds and retarded point spectrum. The covariance operator of the stationary solution is described as a solution of some retarded operator equation.

We study an inverse problem for the two-dimensional random Schrodinger equation (Δ + q + k 2)u = 0. The potential q(x) is assumed to be a Gaussian random function whose covariance operator is a classical pseudodifferential operator. We show that the backscattered field, obtained from a single realization of the random potential q, determines uniquely the principal symbol of the covariance operator of q. The analysis is carried out by combining harmonic and microlocal analysis with stochastic methods.

One-dimensional wave equations with cubic power law perturbed by Q-regular additive space-time random noise are considered. These models describe the displacement of nonlinear strings excited by state-independent random external forces. The presented analysis is based on the representation of its solution in form of Fourier-series expansions along the eigenfunctions of Laplace operator with continuous, Markovian, unique Fourier coefficients (the so-called commutative case). We shall discuss existence and uniqueness of Fourier solutions using energy-type methods based on the construction of Lyapunov-functions. Appropriate truncations and finite-dimensional approximations are presented while exploiting the explicit knowledge on eigenfunctions of related second order differential operators. Moreover, some nonstandard partial-implicit difference methods for their numerical integration are suggested in order to control its energy functional in a dynamically consistent fashion. The generalized energy $\cE$ (sum of kinetic, potential and damping energy) is governed by the linear relation $\E [\varepsilon(t)] = \E [\varepsilon(0)] + b^2 trace (Q) t / 2$ in time $t \ge 0$, where $b$ is the scalar intensity of noise and $Q$ its covariance operator.

Many results in the theory of Gaussian processes rely on the eigenstructure of the covariance operator. However, eigenproblems are notoriously hard to solve explicitly and closed form solutions are known only in a limited number of cases. In this paper we set up a framework for the spectral analysis of the fractional type covariance operators, corresponding to an important family of processes, which includes the fractional Brownian motion and its noise. We obtain accurate asymptotic approximations for the eigenvalues and the eigenfunctions. Our results provide a key to several problems, whose solution is long known in the standard Brownian case, but was missing in the more general fractional setting. This includes computation of the exact limits of $L^2$-small ball probabilities and asymptotic analysis of singularly perturbed integral equations, arising in mathematical physics and applied probability., Comment: several typos fixed

Existence and uniqueness of approximate strong solutions of stochastic infinite-dimensional systems d u = [ A ( t ) u + B ( t , u ) ] d t + G ( t , u ) d W , u ( 0 , ⋅ ) = u 0 ∈ H , t ⩾ 0 with local Lipschitz-continuous, time-depending nonrandom operators A , B and G acting on a separable Hilbert space H are studied. For this purpose, some monotonicity conditions on those operators and an existing U-series expansion of the space–time Wiener process W (U-valued, U ⊆ H , U Hilbert space) with ∑ n = 1 + ∞ α n 2 + ∞ belonging to the trace of related covariance operator Q of W with local noise intensities α n 2 ∈ R 1 as eigenvalues of Q are exploited.

We obtain size estimates for the distribution function of the bilinear Hilbert transform acting on a pair of characteristic functions of sets of finite measure, that yield exponential decay at infinity and blowup near zero to the power −2/3 (modulo some logarithmic factors). These results yield all known Lp bounds for the bilinear Hilbert transform and provide new restricted weak type endpoint estimates on Lp1 × Lp2 when either 1/p1 + 1/p2 = 3/2 or one of p1, p2 is equal to 1. As a consequence of this work we also obtain that the square root of the bilinear Hilbert transform of two characteristic functions is exponentially integrable over any compact set.

In many geoscientific applications, one needs to recover the quantities of interest from indirect observations blurred by colored noise. Such quantities often correspond to the values of bounded linear functionals acting on the solution of some observation equation. For example, various quantities are derived from harmonic coefficients of the Earth’s gravity potential. Each such coefficient is the value of the corresponding linear functional. The goal of this paper is to discuss new means to use information about the noise covariance structure, which allows order-optimal estimation of the functionals of interest and does not involve a covariance operator directly in the estimation process. It is done on the basis of a balancing principle for the choice of the regularization parameter, which is new in geoscientific applications. A number of tests demonstrate its applicability. In particular, we could find appropriate regularization parameters by knowing a small part of the gravitational field on the Earth’s surface with high precision and reconstructing the rest globally by downward continuation from satellite data.

In this paper we discuss the problem of integral representation of analytic functions over a complex Banach space E. We obtain, for a wide class of functions, integral representations of the form f (x) = ∫E′ eγ(x) f1 (γ) dW(γ) and f(x) = ∫E′ 1/1 - γ(x)/∥γ∥ f2(γ) dW(γ), where W is an abstract Wiener measure on E′ and f1, f2 are transformations of f involving the covariance operator of W. © 2004 Elsevier Inc. All rights reserved. Fil:Pinasco, D. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Zalduendo, I. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.

The sharp asymptotics for the entropy-constrained L 2 L^2 -quantiza- tion errors of Gaussian measures on a Hilbert space and in particular, for Gaussian processes is derived. The condition imposed is regular variation of the eigenvalues of the covariance operator.