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We prove the Parseval theorem and the Plancherel Theorem among the analytic Feynman integral and the Fourier Feynman transform and the convolution for the function F : B ν → C in the Class F (B ν) : F ( x → ) = ∫ H exp { i ∑ j = 1 ν (h , x j) ∼ } d μ (h) , μ ∈ M (H) on the Product Abstract Wiener Space (H , B ν , m ν) , where B ν = B × B × ⋯ × B (ν times) and M (H) is the space of a complex valued countably additive measure μ defined on B (H) , the Borel class of a real separable infinite dimensional Hilbert space H. [ABSTRACT FROM AUTHOR]

In this paper, we introduce Wiener integrals with respect to the generalized Hermite process and we prove a non-central limit theorem in which this integral appears as limit. As an application, we investigate the corresponding stochastic differential equations with the generalized Hermite process as a driving noise, we prove the existence and the uniqueness of the solution, and we give a generalization of the Hermite Ornstein–Uhlenbeck process and the Hermite-driving Vasicek process. [ABSTRACT FROM AUTHOR]

In this paper, we establish a rotation theorem for the Wiener integral associated with bounded operator on abstract Wiener spaces. To do this, we introduce a class of the angle preserving operators defined on the abstract Wiener space B. [ABSTRACT FROM AUTHOR]

This article investigates the controllability for neutral stochastic delay functional integro-differential equations driven by a fractional Brownian motion, with Hurst parameter lesser than $ 1/2 $. We employ the theory of resolvent operators developed by [ 10 ] combined with the Banach fixed point theorem to establish sufficient conditions to prove the desired result. [ABSTRACT FROM AUTHOR]

This article explains the history and mathematics of Brownian motion. [ABSTRACT FROM AUTHOR]

We introduce a generalized convolution product (F 8 G)α,β for integral transform Fγ,η for functionals defined on K[0,T] , the space of complex valued continuous functions on [0,T] that vanish at zero. We study some interesting properties of our generalized convolution product and establish various relationships that exist among the generalized convolution product, the integral transform and the first variation for functionals defined on K[0,T] . We also discuss the associativity of the generalized convolution product. [ABSTRACT FROM AUTHOR]

We obtain a maximal inequality for sub-fractional Brownian motion with Hurst indexanalogous to the Burkholder–Davis–Gundy inequality for fractional Brownian motion derived by Novikov and Valkeila [Statist. Probab. Lett.44(1999):47–54] and an integral inequality for Wiener integrals with respect to a sub-fractional Brownian motion with Hurst index. [ABSTRACT FROM AUTHOR]

We investigate the behavior of a first variation of a Fourier transformof a measure of the form :upon the Fourier-Feynman transform and analytic Feynman integral and the convolution, whereand {h1,h2,…,hn} is the orthonormal class of elements inHon the abstract Wiener space (H,B,m). [ABSTRACT FROM PUBLISHER]

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Abstract: We prove that for any second order stochastic process X with stationary increments with continuous paths and continuous variance function, there exists a tempered measure μ (for which we give an explicit expression) related with the domain of the Wiener integral with respect to X as follows: the space of tempered distributions f such that the Fourier transform of f is square integrable with respect to μ is always a dense subset of the domain of the Wiener integral. Moreover, we provide sufficient conditions on μ in order that the domain of the integral is exactly this space of distributions. We apply our results to the fractional Brownian motion. In particular, it is proved that the domain of the Wiener integral with respect to the fractional Brownian motion with Hurst parameter contains distributions that are not given by locally integrable functions, this fact was suggested by Pipiras and Taqqu (2000) in . We have also considered the example of the process given by Ornstein and Uhlenbeck as a model for the position of a Brownian particle. [Copyright &y& Elsevier]

Abstract: Quasi-invariance under translation is established for the σ-finite measure unifying Brownian penalisations, which has been introduced by Najnudel, Roynette and Yor [J. Najnudel, B. Roynette, M. Yor, A remarkable σ-finite measure on related to many Brownian penalisations, C. R. Math. Acad. Sci. Paris 345 (8) (2007) 459–466]. For this purpose, the theory of Wiener integrals for centered Bessel processes, due to Funaki, Hariya and Yor [T. Funaki, Y. Hariya, M. Yor, Wiener integrals for centered Bessel and related processes. II, ALEA Lat. Am. J. Probab. Math. Stat. 1 (2006) 225–240 (electronic)], plays a key role. [Copyright &y& Elsevier]

We investigate the behaviour of the first variation of certain cylinder-type functions: [image omitted] under the Fourier-Feynman transform, where [image omitted] is a Fourier transform of the measure μ on Rn of the form [image omitted] and {α1, α2, ..., αn} is an orthonormal class of L2[0, T] and [image omitted] and [image omitted] are in Rn. And we show that the first variation of [image omitted] behave nicely under Fourier-Feynman transforms. And we show that the Fourier-Feynman transform of the first variation of [image omitted] can be perfectly expressed as the limit of a sequence of Wiener integrals of the first variation on the Wiener space. [ABSTRACT FROM AUTHOR]

For the Wiener integral, one property of inversion is established. This property is used for the construction of nonparametric statistical estimation of the unknown logarithmic derivative for the distribution of random processes, which is observed in Wiener noise. [ABSTRACT FROM AUTHOR]

Abstract: The domain of the Wiener integral with respect to a sub-fractional Brownian motion , , , is characterized. The set is a Hilbert space which contains the class of elementary functions as a dense subset. If , any element of is a function and if , the domain is a space of distributions. [Copyright &y& Elsevier]

We introduce Wiener integrals with respect to the Hermite process and we prove a non-central limit theorem in which this integral appears as limit. As an example, we study a generalization of the fractional Ornstein-Uhlenbeck process. [ABSTRACT FROM AUTHOR]

Abstract: We characterize the domain of the Wiener integral with respect to the fractional Brownian motion of any Hurst parameter on an interval . The domain is the set of restrictions to of the distributions of with support contained in . In the case any element of the domain is given by a function, but in the case this space contains distributions that are not given by functions. The techniques used in the proofs involve distribution theory and Fourier analysis, and allow to study simultaneously both cases and . [Copyright &y& Elsevier]

The purpose of this article is to establish some relationships between L 1 -analytic Fourier-Feynman transforms and Wiener integrals for certain bounded functions F in the Fresnel class ℱ( B ) of the form: F ( x ) = ∫ H exp{i( h , x ) ∼⃒ }dμ( h ), μ ∈ M( H ), on the abstract Wiener space. [ABSTRACT FROM AUTHOR]

A correction to the article "Fourier-Feyman Transforms and Wiener Integrals in the Fresnel Class on Abstract Wiener Spaces" that was published in the April 2005 issue is presented.

We prove that the Wiener integral, the analytic Wiener integral and the analytic Feynman integral of the first variation of F (x) = exp { ∫ 0 T θ (t , x (t)) d t } successfully exist under the certain condition, where θ (t , u) = ∫ R exp { i u v } d σ t (v) is a Fourier–Stieltjes transform of a complex Borel measure σ t ∈ M (R) and M (R) is a set of complex Borel measures defined on R. We will find this condition. Moreover, we prove that the change of scale formula for Wiener integrals about the first variation of F (x) sucessfully holds on the Wiener space. [ABSTRACT FROM AUTHOR]