Banach Space-Valued Ornstein–Uhlenbeck Processes with General Drift Coefficients

We construct Ornstein–Uhlenbeck processes with values in Banach space and with continuous paths. The drift coefficient must only generate a strongly continuous semigroup on the Hilbert space which determines the Brownian motion. We admit arbitrary starting points and consider also invariant measures for the process, generalizing earlier work in many directions. A price for the generality is that sometimes one has to enlarge the phase space but most previously known results are covered.The constructions are based on abstract Wiener space methods, more precisely on images of abstract Wiener spaces under suitable linear transformations of the Cameron–Martin space. The image abstract Wiener measures are then given by stochastic extensions. We present the basic spaces and operators and the most important results on image spaces and stochastic extensions in some detail.     

Evaluation formulas for conditional abstract wiener integrals

In this paper we consider abstract Wiener space version of conditional Wiener integrals and establish formulas for evaluating conditional abstract Wiener integrals for various classes of functions on an abstract Wiener space. We then apply our formulas to evaluate certain Wiener integrals and conditional Wiener and Yeh-Wiener integrals     

Various topologies in the Wiener space and Lévy's stochastic area

Summary In this paper, we observe how Lévy's stochastic area looks when we see it through various topologies in the Wiener space. Our theorem implies that it is quite natural from the viewpoint of topology to define a distinct skeleton of Lévy's stochastic areaS(w) for each distinct topology in the Wiener space, or equivalently, for each distinct abstract Wiener space on which the Wiener measure andS(w) are realized. Thus we cannot determine its intrinsic skeleton in the theory of abstract Wiener spaces.     

Analytic Feynman integrals of transforms of variation of cylinder type functions over Wiener paths in abstract Wiener space

In this paper, we evaluate various analytic Feynman integrals of first variation, conditional first variation, Fourier-Feynman transform and conditional Fourier-Feynman transform of cylinder type functions defined over Wiener paths in abstract Wiener space. We also derive the analytic Feynman integral of the conditional Fourier-Feynman transform for the product of the cylinder type functions which define the functions in a Banach algebra introduced by Yoo, with n linear factors.     

Integral transform and convolution of analytic functionals on abstract Wiener space

Yeh defined a convolution of functionals on classical Wiener space and investigated the relationship between the Fourier-Wiener transforms of functionals in certain classes and the Fourier-Wiener transform of their convolution. Yoo extended Yeh's results to abstract Wiener space. In this paper, we introduce the intergal transform and convolution of analytic functionals on abstract Wiener space. And we establish the relationship between the integral transforms of exponential type of analytic functionals and the integral transform of theor convolution. Also we obtain Parseval's and Plancherel's relations for those functionals from this relationship. The main results of Yeh and Yoo then follow from our results as corollaries.     

On Stein?s method for infinite-dimensional Gaussian approximation in abstract Wiener spaces

In this paper, we generalize Stein?s method to “infinite-variate” normal approximation that is an infinite-dimensional approximation by abstract Wiener measures on a real separable Banach space. We first establish a Stein?s identity for abstract Wiener measures and solve the corresponding Stein?s equation. Then we will present a Gaussian approximation theorem using exchangeable pairs in an infinite-variate context. As an application, we will derive an explicit error bound of Gaussian approximation to the distribution of a sum of independent and identically distributed Banach space-valued random variables based on a Lindeberg-Lévy type limit theorem. In addition, an analogous of Berry-Esséen type estimate for abstract Wiener measures will be obtained.     

Total variation and Cheeger sets in Gauss space

The aim of this paper is to study the isoperimetric problem with fixed volume inside convex sets and other related geometric variational problems in the Gauss space, in both the finite and infinite dimensional case. We first study the finite dimensional case, proving the existence of a maximal Cheeger set which is convex inside any bounded convex set. We also prove the uniqueness and convexity of solutions of the isoperimetric problem with fixed volume inside any convex set. Then we extend these results in the context of the abstract Wiener space, and for that we study the total variation denoising problem in this context.     

Harnack and HWI inequalities on infinite-dimensional spaces

In this paper, the dimensional-free Harnack inequalities are established on infinite-dimensional spaces. More precisely, we establish Harnack inequalities for heat semigroup on based loop group and for Ornstein-Uhlenbeck semigroup on the abstract Wiener space. As an application, we establish the HWI inequality on the abstract Wiener space, which contains three important quantities in one inequality, the relative entropy “H”, Wasserstein distance “W”, and Fisher information “I”.     

A fractional isoperimetric problem in the Wiener space

We introduce a notion of fractional perimeter in an abstract Wiener space and show that half-spaces are the only volume-constrained minimisers.     

Fractional order Sobolev spaces on Wiener space

Summary Fractional order Sobolev spaces are introduced on an abstract Wiener space and Donsker's delta functions are defined as generalized Wiener functionals belonging to Sobolev spaces with negative differentiability indices. By using these notions, the regularity in the sense of Hölder continuity of a class of conditional expectations is obtained.     

Some covariance inequalities in Wiener space

In this work we give an account of some covariance inequalities in abstract Wiener space. An FKG inequality is obtained with positivity and monotonicity being defined in terms of a given cone in the underlying Cameron-Martin space. The last part is dedicated to convex and log-concave functionals, including a proof of the Gaussian conjecture for a particular class of log-concave Wiener functionals.     

The Clark-Ocone formula for vector valued Wiener functionals

The classical representation of random variables as the Itô integral of nonanticipative integrands is extended to include Banach space valued random variables on an abstract Wiener space equipped with a filtration induced by a resolution of the identity on the Cameron-Martin space. The Itô integral is replaced in this case by an extension of the divergence to random operators, and the operators involved in the representation are adapted with respect to this filtration in a suitably defined sense.A complete characterization of measure preserving transformations in Wiener space is presented as an application of this generalized Clark-Ocone formula.     

Absolute continuity of the law of an infinite dimensional Wiener functional with respect to the Wiener probability

Summary In this paper we study conditions ensuring that the law of aC([0, 1])-valued functional defined on an abstract Wiener space is absolutely continuous with respect to the Wiener measure onC([0,1]). These conditions extend those established byP. Malliavin [12, 13] for finite-dimensional Wiener functionals, and those of [15] for Hilbert-valued functionals.     

Tangent Processes on Wiener Space

This paper deals with the study of the Malliavin calculus of Euclidean motions on Wiener space (i.e. transformations induced by general measure-preserving transformations, called “rotations”, and H-valued shifts) and the associated flows on abstract Wiener spaces.     

Potential theory of infinite dimensional Lévy processes

We study the potential theory of a large class of infinite dimensional Lévy processes, including Brownian motion on abstract Wiener spaces. The key result is the construction of compact Lyapunov functions, i.e., excessive functions with compact level sets. Then many techniques from classical potential theory carry over to this infinite dimensional setting. Thus a number of potential theoretic properties and principles can be proved, answering long standing open problems even for the Brownian motion on abstract Wiener space, as, e.g., formulated by R. Carmona in 1980. In particular, we prove the analog of the known result, that the Cameron-Martin space is polar, in the Lévy case and apply the technique of controlled convergence to solve the Dirichlet problem with general (not necessarily continuous) boundary data.     

Convergence of Symmetric Diffusions on Wiener Spaces

In this paper,we study the distorted Ornstein-Uhlenbeck processes associated with given densitieson an abstract Wiener space.We prove that the laws of distorted Ornstein-Uhlenbeck processes converge intotal variation norm if the densities converge in Sobolev space D_2~1.     

Some regularity properties of diffusion processes on abstract Wiener space

Diffusion processes on an abstract Wiener space are constructed from fundamental solutions of second-order parabolic equations with variable coefficients. The transition probabilities of such processes are compared with those of the Wiener process, and continuity of sample paths is established. Several operator semigroups associated with the processes are defined (one locally), and some regularity properties of these semigroups are established.     

Regularities for semilinear stochastic partial differential equations

In this paper, we study the regularities of solutions to semilinear stochastic partial differential equations in general settings, and prove that the solution can be smooth arbitrarily when the data is sufficiently regular. As applications, we also study several classes of semilinear stochastic partial differential equations on abstract Wiener space, complete Riemannian manifold as well as bounded domain in Euclidean space.     

Evaluation Formulas for a Conditional Feynman Integral Over Wiener Paths in Abstract Wiener Space

In this paper, we introduce a simple formula for conditional Wiener integrals over , the space of abstract Wiener space valued continuous functions. Using this formula, we establish various formulas for a conditional Wiener integral and a conditional Feynman integral of functionals on in certain classes which correspond to the classes of functionals on the classical Wiener space introduced by Cameron and Storvick. We also evaluate the conditional Wiener integral and conditional Feynman integral for functionals of the form which are of interest in Feynman integration theories and quantum mechanics.     

Triangulation of stratified fibre bundles

We prove that for any stratified fibre bundle p:A·M (A being the underlying space of an abstract prestratification and M a smooth manifold) and any triangulation of M there exists a triangulation of A such that p becomes linear with respect to these triangulations. In particular, any abstract prestratification is triangulable. As a corollary we obtain that the orbit space of a smooth action of a compact Lie group is triangulable.This paper was written while the author was a visiting professor at the Institute of Mathematics of the University of Genova.