Wiener Space Research Articles

Generalized analytic Feynman integrals via the operators and its applications

In this paper, we introduce a new concept of a generalized analytic Feynman integral combining the bounded linear operators on abstract Wiener space. We then obtain some Feynman integration formulas involving the generalized first variation. These formulas are more generalized forms rather than the formulas studied in previous papers. Finally, we establish a generalized Cameron-Storvick theorem, and give some examples to illustrate the usefulness of our results and formulas.

A rotation of Wiener integral associated with bounded operators on abstract Wiener spaces

A rotation of Wiener integral associated with bounded operators on abstract Wiener spaces

Bergman kernels for Paley–Wiener spaces and Nazarov’s proof of the Bourgain–Milman theorem

We give a general inequality for Bergman kernels of Bergman spaces defined by convex weights in $\C^n$. We also discuss how this can be used in Nazarov's proof of the Bourgain-Milman theorem, as a substitute for H\"ormander's estimates for the $\dbar$-equation.

Monte Carlo construction of cubature on Wiener space

AbstractIn this paper, we investigate application of mathematical optimization to construction of a cubature formula on Wiener space, which is a weak approximation method of stochastic differential equations introduced by Lyons and Victoir (Proc R Soc Lond A 460:169–198, 2004). After giving a brief review on the cubature theory on Wiener space, we show that a cubature formula of general dimension and degree can be obtained through a Monte Carlo sampling and linear programming. This paper also includes an extension of stochastic Tchakaloff’s theorem, which technically yields the proof of our primary result.

Cameron–Storvick theorem associated with Gaussian paths on function space

The purpose of this paper is to provide a more general Cameron-Storvick theorem for the generalized analytic Feynman integral associated with Gaussian process $\mathcal Z_k$ on a very general Wiener space $C_{a,b}[0,T]$. The general Wiener space $C_{a,b}[0,T]$ can be considered as the set of all continuous sample paths of the generalized Brownian motion process determined by continuous functions $a(t)$ and $b(t)$ on $[0,T]$. As an interesting application, we apply this theorem to evaluate the generalized analytic Feynman integral of certain monomials in terms of Paley-Wiener-Zygmund stochastic integrals.

Conjugate Phase Retrieval in Paley–Wiener Space

We consider the problem of conjugate phase retrieval in Paley–Wiener space \(PW_{\pi }\). The goal of conjugate phase retrieval is to recover a signal f from the magnitudes of linear measurements up to unknown phase factor and unknown conjugate, meaning f(t) and \(\overline{f(t)}\) are not necessarily distinguishable from the available data. We show that conjugate phase retrieval can be accomplished in \(PW_{\pi }\) by sampling only on the real line by using structured convolutions. We also show that conjugate phase retrieval can be accomplished in \(PW_{\pi }\) by sampling both f and \(f^{\prime }\) only on the real line. Moreover, we demonstrate experimentally that the Gerchberg–Saxton method of alternating projections can accomplish the reconstruction from vectors that do conjugate phase retrieval in finite dimensional spaces. Finally, we show that generically, conjugate phase retrieval can be accomplished by sampling at three times the Nyquist rate, whereas phase retrieval requires sampling at four times the Nyquist rate.

Stochastic integrals and Brownian motion on abstract nilpotent Lie groups

We construct a class of iterated stochastic integrals with respect to Brownian motion on an abstract Wiener space which allows for the definition of Brownian motions on a general class of infinite-dimensional nilpotent Lie groups based on abstract Wiener spaces. We then prove that a Cameron--Martin type quasi-invariance result holds for the associated heat kernel measures in the non-degenerate case, and give estimates on the associated Radon--Nikodym derivative. We also prove that a log Sobolev estimate holds in this setting.

Integrability on the Abstract Wiener Space of Double Sequences and Fernique Theorem

The integrability of a function defined on the abstract Wiener space of double Fourier coefficients is explored. The abstract Wiener space is also a Hilbert space. We define an orthonormal system of the Hilbert space to establish a measure and integration on the abstract Wiener space. We examine the integrability of a function e α · 2 defined on the abstract Wiener space for Fernique theorem. With respect to the abstract Wiener measure, the integral of the function turns out to be convergent for α < 1 / 2 . The result provides a wider choice of the constant α than that of Fernique.

Bilateral Laplace–Feynman transforms on abstract Wiener spaces

ABSTRACT In this paper, an extension of the bilateral Laplace transform on infinite-dimensional Banach spaces is introduced. In order to develop the concept of an analytic bilateral Laplace–Feynman transform (BLFT) of functionals on abstract Wiener spaces, we complete the following objectives: we first establish the existence of the analytic BLFT of certain bounded functionals on B. We next extend the operational properties (shifting properties) of the bilateral Laplace transform of functions on Euclidean spaces to the cases of the BLFT of functionals on abstract Wiener spaces. We also provide a convolution product (CP) corresponding to the BLFT and establish a relationship between the BLFT and the CP. We finally provide a representation of an inverse transform of the BLFT. Specifically, we establish a representation of the inverse transform for the BLFT having special parameter via the concept of the analytic Fourier–Feynman transform (FFT).

Approximation theorems on graphs

Analysis of functions on combinatorial graphs is an emerging field attracting more and more attention. In this paper, we study the approximation of functions defined on combinatorial graphs by functions in Paley–Wiener spaces. First, we use a family of graph translation operators to define the modulus of smoothness, which has several properties similar to their counterparts in the classical approximation theory. Next, we establish Jackson’s and Bernstein’s inequalities for functions defined on graphs. Finally, we provide an estimation on the decay of graph Fourier coefficients in terms of the modulus of smoothness. These results lead to a theory of approximation of functions on combinatorial graphs and have potential applications to filtering, denoising, data dimension reduction, image processing and learning theory.

Decoupling on the Wiener Space, Related Besov Spaces, and Applications to BSDEs

We introduce a decoupling method on the Wiener space to define a wide class of anisotropic Besov spaces. The decoupling method is based on a general distributional approach and not restricted to the Wiener space. The class of Besov spaces we introduce contains the traditional isotropic Besov spaces obtained by the real interpolation method, but also new spaces that are designed to investigate backwards stochastic differential equations (BSDEs). As examples we discuss the Besov regularity (in the sense of our spaces) of forward diffusions and local times. It is shown that among our newly introduced Besov spaces there are spaces that characterize quantitative properties of directional derivatives in the Malliavin sense without computing or accessing these Malliavin derivatives explicitly. Regarding BSDEs, we deduce regularity properties of the solution processes from the Besov regularity of the initial data, in particular upper bounds for their L p L_p -variation, where the generator might be of quadratic type and where no structural assumptions, for example in terms of a forward diffusion, are assumed. As an example we treat sub-quadratic BSDEs with unbounded terminal conditions. Among other tools, we use methods from harmonic analysis. As a by-product, we improve the asymptotic behaviour of the multiplicative constant in a generalized Fefferman inequality and verify the optimality of the bound we established.

Multivariate Normal Approximation on the Wiener Space: New Bounds in the Convex Distance

AbstractWe establish explicit bounds on the convex distance between the distribution of a vector of smooth functionals of a Gaussian field and that of a normal vector with a positive-definite covariance matrix. Our bounds are commensurate to the ones obtained by Nourdin et al. (Ann Inst Henri Poincaré Probab Stat 46(1):45–58, 2010) for the (smoother) 1-Wasserstein distance, and do not involve any additional logarithmic factor. One of the main tools exploited in our work is a recursive estimate on the convex distance recently obtained by Schulte and Yukich (Electron J Probab 24(130):1–42, 2019). We illustrate our abstract results in two different situations: (i) we prove a quantitative multivariate fourth moment theorem for vectors of multiple Wiener–Itô integrals, and (ii) we characterize the rate of convergence for the finite-dimensional distributions in the functional Breuer–Major theorem.

Information Projection on Banach Spaces with Applications to State Independent KL-Weighted Optimal Control

This paper studies constrained information projections on Banach spaces with respect to a Gaussian reference measure. Specifically our interest lies in characterizing projections of the reference measure, with respect to the KL-divergence, onto sets of measures corresponding to changes in the mean (or shift measures). As our main result, we give a portmanteau theorem that characterizes the relationship among several different formulations of this problem. In the general setting of Gaussian measures on a Banach space, we show that this information projection problem is equivalent to minimization of a certain Onsager–Machlup (OM) function with respect to an associated stochastic process. We then construct several reformulations in the more specific setting of classical Wiener space. First, we show that KL-weighted optimization over shift measures can also be expressed in terms of an OM function for an associated stochastic process that we are able to characterize. Next, we show how to encode the feasible set of shift measures through an explicit functional constraint by constructing an appropriate penalty function. Finally, we express our information projection problem as a calculus of variations problem, which suggests a solution procedure via the Euler–Lagrange equation. We work out the details of these reformulations for several specific examples.

Evaluation Formula and Approximation for Wiener Integrals via the Fourier-Type Functional

In order to calculate the Wiener integrals for functionals on Wiener space, one can usually apply the change of variable theorem. But, there are many functionals that are difficult or impossible to calculate even when using the change of variable formula. In order to solve this problem, we establish an evaluation formula via the Fourier-type functionals on Wiener space. We then present various examples to which our evaluation formula can be applied and with the corresponding numerical approximations.

Limiting behavior of large correlated Wishart matrices with chaotic entries

We study the fluctuations, as d,n→∞, of the Wishart matrix Wn,d=1dXn,dXn,dT associated to a n×d random matrix Xn,d with non-Gaussian entries. We analyze the limiting behavior in distribution of Wn,d in two situations: when the entries of Xn,d are independent elements of a Wiener chaos of arbitrary order and when the entries are partially correlated and belong to the second Wiener chaos. In the first case, we show that the (suitably normalized) Wishart matrix converges in distribution to a Gaussian matrix while in the correlated case, we obtain its convergence in law to a diagonal non-Gaussian matrix. In both cases, we derive the rate of convergence in the Wasserstein distance via Malliavin calculus and analysis on Wiener space.

The Hilbert Space of Double Fourier Coefficients for an Abstract Wiener Space

Fourier series is a well-established subject and widely applied in various fields. However, there is much less work on double Fourier coefficients in relation to spaces of general double sequences. We understand the space of double Fourier coefficients as an abstract space of sequences and examine relationships to spaces of general double sequences: p-power summable sequences for p = 1, 2, and the Hilbert space of double sequences. Using uniform convergence in the sense of a Cesàro mean, we verify the inclusion relationships between the four spaces of double sequences; they are nested as proper subsets. The completions of two spaces of them are found to be identical and equal to the largest one. We prove that the two-parameter Wiener space is isomorphic to the space of Cesàro means associated with double Fourier coefficients. Furthermore, we establish that the Hilbert space of double sequence is an abstract Wiener space. We think that the relationships of sequence spaces verified at an intermediate stage in this paper will provide a basis for the structures of those spaces and expect to be developed further as in the spaces of single-indexed sequences.

On integration by parts formula on open convex sets in Wiener spaces

In Euclidean space, it is well known that any integration by parts formula for a set of finite perimeter \(\Omega \) is expressed by the integration with respect to a measure \(P(\Omega ,\cdot )\) which is equivalent to the one-codimensional Hausdorff measure restricted to the reduced boundary of \(\Omega \). The same result has been proved in an abstract Wiener space, typically an infinite-dimensional space, where the surface measure considered is the one-codimensional spherical Hausdorff–Gauss measure \(\mathscr {S}^{\infty -1}\) restricted to the measure-theoretic boundary of \(\Omega \). In this paper, we consider an open convex set \(\Omega \) and we provide an explicit formula for the density of \(P(\Omega ,\cdot )\) with respect to \(\mathscr {S}^{\infty -1}\). In particular, the density can be written in terms of the Minkowski functional \(\mathfrak {p}\) of \(\Omega \) with respect to an inner point of \(\Omega \). As a consequence, we obtain an integration by parts formula for open convex sets in Wiener spaces.

Well-posedness of the water-wave with viscosity problem

In this paper we study the motion of a surface gravity wave with viscosity. In particular we prove two well-posedness results. On the one hand, we establish the local solvability in Sobolev spaces for arbitrary dissipation. On the other hand, we establish the global well-posedness in Wiener spaces for a sufficiently large viscosity. These results are the first rigorous proofs of well-posedness for the Dias, Dyachenko & Zakharov system (Physics Letters A 2008) modeling gravity waves with viscosity when surface tension is not taken into account.

Fractional Paley\u2013Wiener and Bernstein spaces

We introduce and study a family of spaces of entire functions in one variable that generalise the classical Paley–Wiener and Bernstein spaces. Namely, we consider entire functions of exponential type a whose restriction to the real line belongs to the homogeneous Sobolev space dot{W}^{s,p} and we call these spaces fractional Paley–Wiener if p=2 and fractional Bernstein spaces if pin (1,infty ), that we denote by PW^s_a and {mathcal {B}}^{s,p}_a, respectively. For these spaces we provide a Paley–Wiener type characterization, we remark some facts about the sampling problem in the Hilbert setting and prove generalizations of the classical Bernstein and Plancherel–Pólya inequalities. We conclude by discussing a number of open questions.

Stochastic differential equations with singular coefficients on the straight line

Consider the following stochastic differential equation (SDE): Xt=x+∫0tb(s,Xs)ds+∫0tσ(s,Xs)dBs,0≤t≤T,x∈R,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ X_{t}=x+ \\int _{0}^{t}b(s,X_{s})\\,ds+ \\int _{0}^{t}\\sigma (s,X_{s}) \\,dB_{s}, \\quad 0\\leq t\\leq T, x\\in \\mathbb{R}, $$\\end{document} where {B_{s}}_{0leq sleq T} is a 1-dimensional standard Brownian motion on [0,T]. Suppose that qin (1,infty ], pin (1,infty ), b=b_{1}+b_{2}, b_{1}in L^{q}(0,T;L^{p}(mathbb{R})) such that 1/p+2/q<1 and b_{2} is bounded measurable, with sigma in L^{infty }(0,T;{mathcal{C}}_{u}(mathbb{R})) there being a real number delta >0 such that sigma ^{2}geq delta . Then there exists a weak solution to the above equation. Moreover, (i) if sigma in mathcal{C}([0,T];mathcal{C}_{u}(mathbb{R})), all weak solutions have the same probability law on 1-dimensional classical Wiener space on [0,T] and there is a density associated with the above SDE; (ii) if b_{2}=0, pin [2,infty ) and sigma in L^{2}(0,T;{mathcal{C}}_{b}^{1/2}({mathbb{R}})), the pathwise uniqueness holds.