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.     

Analytic functionals on the Wiener space

We define the analyticity of Wiener functionals and study its properties and applications to oscillatory Wiener functionals. Project supported by the National Natural Science Foundation of China.     

Dimension-Independent Estimates on the Densities of Wiener Functionals via the Log-Sobolev Inequality

Using the log-Sobolev inequality, we shall present in this note some estimates on the density of finite dimensional non-degenerate Wiener functionals which are independent on the dimension. We shall take the Gaussian measure as the reference measure, contrary to the customary choice of Lebesgue measure in the literature. As an application, we show that the limit in probability of a uniformly bounded sequence of non-degenerate Wiener functionals has a density with respect to the Gaussian measure.     

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.     

Deviation inequalities for quadratic Wiener functionals and moderate deviations for parameter estimators

We study deviation inequalities for some quadratic Wiener functionals and moderate deviations for parameter estimators in a linear stochastic differential equation model. Firstly, we give some estimates for Laplace integrals of the quadratic Wiener functionals by calculating the eigenvalues of the associated Hilbert-Schmidt operators. Then applying the estimates, we establish deviation inequalities for the quadratic functionals and moderate deviation principles for the parameter estimators.     

The wavelet transform for Wiener functionals and some applications

The wavelet transform is defined for Wiener functionals. We characterize global and local regularities of Wiener functionals and we give a criterion for the existence and regularity of densities. Such a criterion is applied to diffusion processes and to the solutions to backward stochastic differential equations.     

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.     

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.     

Second order Poincaré inequalities and CLTs on Wiener space

We prove infinite-dimensional second order Poincaré inequalities on Wiener space, thus closing a circle of ideas linking limit theorems for functionals of Gaussian fields, Stein's method and Malliavin calculus. We provide two applications: (i) to a new “second order” characterization of CLTs on a fixed Wiener chaos, and (ii) to linear functionals of Gaussian-subordinated fields.     

Smoothness of Densities of Generalized Locally Non-Degenerate Wiener Functionals

Several criteria for existence of smooth densities of Wiener functionals are known in the framework of Malliavin calculus. In this article, we introduce the notion of generalized locally non-degenerate Wiener functionals and prove that they possess smooth densities. The result presented here unifies the earlier works by Shigekawa and Florit-Nualart. As an application, we prove that the law of the strong solution to a stochastic differential equation driven by Brownian motion admits a smooth density without an assumption of Lipschitz continuity for dispersion coefficients.     

On the subexponentiality of the ridgelet transform

We show that we can consider the ridgelet transform for Wiener functionals as a subexponential random variable. We give an application of this result to random walks.     

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.     

Upper and lower functions of plane Brownian angles

One of the most rapidly growing fields in the theory of random processes is asymptotics of multidimensional Wiener functionals, including such functionals as winding angles or radius-vectors of planar processes. We present a new method yielding a large number of results on upper and lower functions for such functionals. Bibliography: 23 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 228, 1996, pp. 111–134.     

Poisson discretizations of Wiener functionals and Malliavin operators with Wasserstein estimates

This article proposes a global, chaos-based procedure for the discretization of functionals of Brownian motion into functionals of a Poisson process with intensity $\lambda >0$. Under this discretization we study the weak convergence, as the intensity of the underlying Poisson process goes to infinity, of Poisson functionals and their corresponding Malliavin-type derivatives to their Wiener counterparts. In addition, we derive a convergence rate of $O\left({\lambda }^{?1∕4}\right)$ for the Poisson discretization of Wiener functionals by combining the multivariate Chen–Stein method with the Malliavin calculus. Our proposed sufficient condition for establishing the mentioned convergence rate involves the kernel functions in the Wiener chaos, yet we provide examples, especially the discretization of some common path dependent Wiener functionals, to which our results apply without committing the explicit computations of such kernels. To the best our knowledge, these are the first results in the literature on the universal convergence rate of a global discretization of general Wiener functionals.     

广义维纳泛函,白噪声分布及实Schwartz广义函数空间上的分布

在实Ｓｃｈｗａｒｔｚ广义函数空间上，证明了复值广义维纳泛函，由Ｋｏｎｄｒａｔｅｖ－Ｓｔｒｅｉｔ及Ｈｉｄａ构造的复值白噪声分布都是由Ｋｈｒｅｎｎｉｋｏｖ构造的分布。利用上述结果进而证明了，一类无穷维伪微分算子是由复值广义维纳泛函空间上的连续线性算子族扩张而成。更进一步，还证明了由Ｋｈｒｅｎｎｉｋｏｖ构造的关于分布的试验函数空间是关于白噪声泛函的Ｍｅｙｅｒ－Ｙａｎ试验函数空间的子空间。     

La transformée en ridelettes pour les fonctionnelles de Wiener

The ridgelet transform is extended to Wiener functionals. Even if the reconstruction property is not satisfied, we still approximate a Wiener functional with the help of its transform and we give an application to its regularity.     

A recurrence formula with respect to the Cameron–Storvick type theorem of the 𝒯-transform

ABSTRACT

In this paper, we define a transform which has the kernel in its definition and a concept of derivative for functionals on Wiener space. We then establish some results and formulas for the transforms of functionals on Wiener space. We also establish the Cameron–Storvick type theorem for the transform. Finally, we obtain the recurrence formula for the transforms to evaluate formulas involving the multi-dimensional derivative.     

On Wiener functionals of order 2 associated with soliton solutions of the KdV equation

The eigenvalues and eigenvectors of the Hilbert-Schmidt operators corresponding to the Wiener functionals of order 2, which give a rise of soliton solutions of the KdV equation, are determined. Two explicit expressions of the stochastic oscillatory integral with such Wiener functional as phase function are given; one is of infinite product type and the other is of Lévy's formula type. As an application, the asymptotic behavior of the stochastic oscillatory integral will be discussed.     

Dimension Free and Infinite Variance Tail Estimates on Poisson Space

Concentration inequalities are obtained on Poisson space, for random functionals with finite or infinite variance. In particular, dimension free tail estimates and exponential integrability results are given for the Euclidean norm of vectors of independent functionals. In the finite variance case these results are applied to infinitely divisible random variables such as quadratic Wiener functionals, including Lévy’s stochastic area and the square norm of Brownian paths. In the infinite variance case, various tail estimates such as stable ones are also presented.      

Normal convergence using Malliavin calculus with applications and examples

We prove the chain rule in the more general framework of the Wiener–Poisson space, allowing us to obtain the so-called Nourdin–Peccati bound. From this bound, we obtain a second-order Poincaré-type inequality that is useful in terms of computations. For completeness we survey these results on the Wiener space, the Poisson space, and the Wiener–Poisson space. We also give several applications to central limit theorems with relevant examples: linear functionals of Gaussian subordinated fields (where the subordinated field can be processes like fractional Brownian motion or the solution of the Ornstein–Uhlenbeck SDE driven by fractional Brownian motion), Poisson functionals in the first Poisson chaos restricted to infinitely many “small” jumps (particularly fractional Lévy processes), and the product of two Ornstein–Uhlenbeck processes (one in the Wiener space and the other in the Poisson space). We also obtain bounds for their rate of convergence to normality.