Wiener Space Research Articles

A class of infinite dimensional stochastic processes with unbounded diffusion

The paper studies Dirichlet forms on the classical Wiener space and the Wiener space over non-compact complete Riemannian manifolds. The diffusion operator is almost everywhere an unbounded operator on the Cameron–Martin space. In particular, it is shown that under a class of changes of the reference measure, quasi-regularity of the form is preserved. We also show that under these changes of the reference measure, derivative and divergence are closable with certain closable inverses. We first treat the case of the classical Wiener space and then we transfer the results to the Wiener space over a Riemannian manifold.

ADMIXABLE OPERATORS AND A TRANSFORM SEMIGROUP ON ABSTRACT WIENER SPACE

The purpose of this paper is first of all to investigate the behavior of admixable operators on the product of abstract Wiener spaces and secondly to examine transform semigroups which consist of admix-Wiener transforms on abstract Wiener spaces.

Series expansions of the transform with respect to the Gaussian process

In this paper, we consider the Fourier-type functionals introduced in [Chung HS, Tuan VK. Fourier-type functionals on Wiener space. Bull Korean Math Soc. 2012;49:609–619] and used in [Lee IY, Chung HS, Chang SJ. Series expansions of the analytic Feynman integral for the Fourier-type functional. J Korean Soc Math Educ Ser B. 2012;19:87–102]. We first establish the existence of the transform with respect to the Gaussian process for the Fourier-type functionals. We then obtain the various relationships for the transform with respect to the Gaussian process of the Fourier-type functionals which involved the ⋄-product and in addition, establish various integration formulas.

Elliptic PDEs with constant coefficients on convex polyhedra via the unified method

We provide a new method to study the classical Dirichlet problem for constant coefficient second order elliptic PDEs on convex polyhedrons. Our approach is heavily motivated by Fokas' unified method for boundary value problems, and can be interpreted as the Fourier analogue to the classical boundary integral equations. The central object in this approach is the global relation: an integral equation which couples the known boundary data and the unknown boundary values. This integral equation depends holomorphically on two complex parameters, and the resulting analysis takes place on a Banach space of complex analytic functions closely related to the classical Paley–Wiener space. We write the global relation in the form of an operator equation and show that the analysis can be reduced to the case of Laplace's equation, from which the more general problem turns out to be a compact perturbation. We give a new integral representation to the solution to the underlying boundary value problem which serves as a concrete realisation of the fundamental principle of Ehrenpreis for all constant coefficient elliptic PDEs on convex polyhedra.

An analytic bilateral Laplace–Feynman transform on Hilbert space

In this paper, we examine the analytic bilateral Laplace–Feynman transform (BLFT) for functions on the Hilbert space H. We then proceed to establish a relationship between the analytic BLFT on H and the analytic Fourier–Feynman transform on the abstract Wiener space B.

A fractional Brownian field indexed by [formula omitted] and a varying Hurst parameter

Using structures of Abstract Wiener Spaces, we define a fractional Brownian field indexed by a product space (0,1/2]×L2(T,m), (T,m) a separable measure space, where the first coordinate corresponds to the Hurst parameter of fractional Brownian motion. This field encompasses a large class of existing fractional Brownian processes, such as Lévy fractional Brownian motions and multiparameter fractional Brownian motions, and provides a setup for new ones. We prove that it has satisfactory incremental variance in both coordinates and derive certain continuity and Hölder regularity properties in relation with metric entropy. Also, a sharp estimate of the small ball probabilities is provided, generalizing a result on Lévy fractional Brownian motion. Then, we apply these general results to multiparameter and set-indexed processes, proving the existence of processes with prescribed local Hölder regularity on general indexing collections.

Exponential bases, Paley–Wiener spaces and applications

We investigate the connection between translation bases for Paley–Wiener spaces and exponential Fourier bases for a domain. We apply these results to the characterization of vector-valued time–frequency translations of a Paley–Wiener “window” signal.

Sampling, interpolation and Riesz bases in small Fock spaces

We give a complete description of Riesz bases of reproducing kernels in small Fock spaces. This characterization is in the spirit of the well known Kadets–Ingham 1/4 theorem for Paley–Wiener spaces. Contrary to the situation in Paley–Wiener spaces, a link can be established between Riesz bases in the Hilbert case and corresponding complete interpolating sequences in small Fock spaces with associated uniform norm. These results allow to show that if a sequence has a density strictly different from the critical one then either it can be completed or reduced to a complete interpolating sequence. In particular, this allows to give necessary and sufficient conditions for interpolation or sampling in terms of densities.

Lagrangian flows driven by $$BV$$ fields in Wiener spaces

We establish the renormalization property for essentially bounded solutions of the continuity equation associated to bounded variation (\(BV\)) fields in Wiener spaces, with values in the associated Cameron-Martin space; thus obtaining, by standard arguments, new uniqueness and stability results for correspondent Lagrangian flows.

Recovery of bivariate band-limited functions using scattered translates of the Poisson kernel

This paper continues the study of interpolation operators on scattered data. We introduce the Poisson interpolation operator and prove various properties of this operator. The main result concerns functions whose Fourier transforms are concentrated near the origin, specifically functions belonging to the Paley–Wiener space PWBβ. We show that one may recover these functions from their samples on a complete interpolating sequence for [−δ,δ]2 by using the Poisson interpolation operator, provided that 0<β<(3−8)δ.

The Zero-Removing Property in Hilbert Spaces of Entire Functions Arising in Sampling Theory

In the topic of sampling in reproducing kernel Hilbert spaces, sampling in Paley–Wiener spaces is the paradigmatic example. A natural generalization of Paley–Wiener spaces is obtained by substituting the Fourier kernel with an analytic Hilbert-space-valued kernel K. Thus we obtain a reproducing kernel Hilbert space \({\mathcal{H}_{K}}\) of entire functions in which the Kramer property allows to prove a sampling theorem. A necessary and sufficient condition ensuring that this sampling formula can be written as a Lagrange-type interpolation series concerns the stability under removal of a finite number of zeros of the functions belonging to the space \({\mathcal{H}_{K}}\); this is the so-called zero-removing property. This work is devoted to the study of the zero-removing property in \({\mathcal{H}_{K}}\) spaces, regardless of the Kramer property, revealing its connections with other mathematical fields.

Nourdin–Peccati analysis on Wiener and Wiener–Poisson space for general distributions

Given a reference random variable, we study the solution of its Stein equation and obtain universal bounds on its first and second derivatives. We then extend the analysis of Nourdin and Peccati by bounding the Fortet–Mourier and Wasserstein distances from more general random variables such as members of the Exponential and Pearson families. Using these results, we obtain non-central limit theorems, generalizing the ideas applied to their analysis of convergence to Normal random variables. We do these in both Wiener space and the more general Wiener–Poisson space. In the former space, we study conditions for convergence under several particular cases and characterize when two random variables have the same distribution. In the latter space we give sufficient conditions for a sequence of multiple (Wiener–Poisson) integrals to converge to a Normal random variable.

Second order PDEs with Dirichlet white noise boundary conditions

In this paper we study the Poisson and heat equations on bounded and unbounded domains with smooth boundary with random Dirichlet boundary conditions. The main novelty of this work is a convenient framework for the analysis of such equations excited by the white in time and/or space noise on the boundary. Our approach allows us to show the existence and uniqueness of weak solutions in the space of distributions. Then we prove that the solutions can be identified as smooth functions inside the domain, and finally the rate of their blow up at the boundary is estimated. A large class of noises including Wiener and fractional Wiener space time white noise, homogeneous noise and Levy noise is considered.

The average errors for linear combinations of Bernstein–Kantorovich operators on the Wiener space

In this paper, we discuss the average errors of function approximation by linear combinations of Bernstein–Kantorovich operators. The strongly asymptotic orders for the average errors of these operators sequence are determined on the Wiener space.

Average errors for Kantorovitch operators on r-fold integrated Wiener space

For weighted approximation in Lp-norm, we determine strongly asymptotic orders for simultaneous approximation average errors of sequence of Kantorovitch operators on r-fold integrated Wiener space. These results show the saturation properties of the Kantorovitch operators in the average case setting.

Frame properties of shifts of prolate spheroidal wave functions

We provide conditions on a shift parameter and number of basic prolate spheroidal wave functions with a fixed bandwidth and time concentrated to a fixed duration such that the shifts of the basic prolates form a frame or a Riesz basis for the Paley–Wiener space consisting of all square integrable functions with the given bandlimit.

Variational calculation of Laplace transforms via entropy on Wiener space and applications

Let (W,H,μ) be the classical Wiener space where H is the Cameron–Martin space which consists of the primitives of the elements of L2([0,1],dt)⊗Rd. We denote by La2(μ,H) the equivalence classes w.r.t. dt×dμ whose Lebesgue densities s→u˙(s,w) are almost surely adapted to the canonical Brownian filtration. If f is a Wiener functional s.t. 1E[e−f]e−fdμ is of finite relative entropy w.r.t. μ, we prove thatJ⋆=inf⁡(Eμ[f∘U+12|u|H2]:u∈La2(μ,H))≥−log⁡Eμ[e−f]=inf⁡(∫Wfdγ+H(γ|μ):γ∈P(W)) where P(W) is the set of probability measures on (W,B(W)) and H(γ|μ) is the relative entropy of γ w.r.t. μ. We call f a tamed functional if the inequality above can be replaced with equality. We characterize the class of tamed functionals, which is much larger than the set of essentially bounded Wiener functionals. We show that for a tamed functional the minimization problem of l.h.s. has a solution u0 if and only if U0=IW+u0 is almost surely invertible anddU0μdμ=e−fEμ[e−f] and then u0 is unique. To do this, we prove the theorem which says that the relative entropy of U0μ is equal to the energy of u0 if and only if it has a μ-a.s. left inverse. We use these results to prove the strong existence of the solutions of stochastic differential equations with singular (functional) drifts and also to prove the non-existence of strong solutions of some stochastic differential equations.

The Gaussian Radon transform in classical Wiener space

We study the Gaussian Radon transform in the classical Wiener space of Brownian motion. We determine explicit formulas for transforms of Brownian functionals specified by stochastic integrals. A Fock space decom- position is also established for Gaussian measure conditioned to closed affine subspaces in Hilbert spaces.

Relationships for modified generalized integral transforms, modified convolution products and first variations on function space

In this paper we define a modified generalized integral transform (MGIT) and a modified convolution product (MCP) for functionals on function space. We then introduce classes of functionals which are dense in L2(Ca, b[0, T]) and establish various relationships between the MGIT and the MCP involving the first variation. Note that all results in [Chang SJ, Chung HS, Skoug D. Integral transforms of functionals in L2(Ca, b[0, T]). J Fourier Anal Appl. 2009;15:441–462; Chang SJ, Chung HS, Skoug D. Some basic relationships among transforms, convolution products, first variations and inverse transforms. Central Eur J Math. 2013;11:538–551; Chang KS, Kim BS, Yoo I. Integral transform and convolution of analytic functionals on abstract Wiener space. Numer Funct Anal Optim. 2000;21:97–105; Im MK, Ji UC, Park YJ. Relations among the first variation, the convolutions and the generalized Fourier–Gauss transforms. Bull Korean Math Soc. 2011;48:291–302; Lee Y-J. Integral transforms of analytic functions on abstract Wiener spaces. J Funct Anal. 1982;47:153–164; Kim BS, Skoug D. Integral transforms of functionals in L2(C0[0, T]). Rocky Mountain J Math. 2003;33:1379–1393] and [Chung HS, Tuan VK. Generalized integral transforms and convolution products on function space. Integral Transforms Spec Funct. 2011;22:573–586] are then corollaries of the results and formulas established in this paper.

RELATIONSHIP BETWEEN THE WIENER INTEGRAL AND THE ANALYTIC FEYNMAN INTEGRAL OF CYLINDER FUNCTION

Cameron and Storvick discovered a change of scale for- mula for Wiener integral of functionals in a Banach algebra S on classical Wiener space. We express the analytic Feynman integral of cylinder function as a limit of Wiener integrals. Moreover we obtain the same change of scale formula as Cameron and Storvick's result for Wiener integral of cylinder function. Our result cover a restricted version of the change of scale formula by Kim.