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Jae Gil Choi

Publications and source records attributed to Jae Gil Choi.

11 recordsLinked to original sources
arXiv2021

Analytic operator-valued generalized Feynman integral on function space

In this paper an analytic operator-valued generalized Feynman integral was studied on a very general Wiener space $C_{a,b}[0,T]$. The general Wiener space $C_{a,b}[0,T]$ is a function space which is induced by the generalized Brownian motion process associated with continuous functions $a$ and $b$. The structure of the analytic operator-valued generalized Feynman integral is suggested and the existence of the analytic operator-valued generalized Feynman integral is investigated as an operator from $L^1(\mathbb R, ν_{δ,a})$ to $L^{\infty}(\mathbb R)$ where $ν_{δ,a}$ is a $σ$-finite measure on $\mathbb R$ given by \[ dν_{δ,a}=\exp\{δ\mathrm{Var}(a)u^2\} du, \] where $δ>0$ and $\mathrm{Var}(a)$ denotes the total variation of the mean function $a$ of the generalized Brownian motion process. It turns out in this paper that the analytic operator-valued generalized Feynman integrals of functionals defined by the stochastic Fourier--Stieltjes transform of complex measures on the infinite dimensional Hilbert space $C_{a,b}'[0,T]$ are elements of the linear space \[ \bigcap_{δ>0} \mathcal L( L^1(\mathbb R,ν_{δ,a}),L^{\infty}(\mathbb R)). \]

math.PR↗
arXiv2021

Fubini theorems for analytic Yeh-Feynman integrals associated with Gaussian processes with applications

In this paper we study an analytic Yeh--Feynman integral and an analytic Yeh--Fourier--Feynman transform associated with Gaussian processes. Fubini theorems involving the generalized analytic Yeh--Feynman integrals are established. The Fubini theorems investigated in this paper are to express the iterated generalized Yeh--Feynman integrals associated with Gaussian processes as a single generalized Yeh--Feynman integral. Using our Fubini theorems, we next examined fundamental relationships (with extended versions) between generalized Yeh--Fourier--Feynman transforms and convolution products (with respect to Gaussian processes) of functionals on Yeh--Wiener space.

math.FA↗
arXiv2021

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.

math.PR↗
arXiv2020

A Cameron-Storvick type theorem on $C_{a,b}^2[0,T]$ with applications

The purpose of this paper is to establish a very general Cameron-Storvick theorem involving the generalized analytic Feynman integral of functionals on the product function space $C_{a,b}^2[0,T]$. The function space $C_{a,b}[0,T]$ can be induced by the generalized Brownian motion process associated with continuous functions $a$ and $b$. To do this we first introduce the class $\mathcal F_{A_1,A_2}^{\,\,a,b}$ of functionals on $C_{a,b}^2[0,T]$ which is a generalization of the Kallianpur and Bromley Fresnel class $\mathcal F_{A_1,A_2}$. We then proceed to establish a Cameron-Storvick type theorem on the product function space $C_{a,b}^2[0,T]$. Finally we use our Cameron--Storvick type theorem to obtain several meaningful results and examples.

math.FA↗
arXiv2019

Generalized Fourier--Feynman transforms and generalized convolution products on Wiener space II

The purpose of this article is to present the second type fundamental relationship between the generalized Fourier--Feynman transform and the generalized convolution product on Wiener space. The relationships in this article are also natural extensions (to the case on an infinite dimensional Banach space) of the structure which exists between the Fourier transform and the convolution of functions on Euclidean spaces.

math.FA↗
arXiv2019

Algebraic structure of the $L_2$ analytic Fourier-Feynman transform associated with Gaussian processes on Wiener space

In this paper we study algebraic structures of the classes of the $L_2$ analytic Fourier-Feynman transforms on Wiener space. To do this we first develop several rotation properties of the generalized Wiener integral associated with Gaussian processes. We then proceed to analyze the $L_2$ analytic Fourier-Feynman transforms associated with Gaussian processes. Our results show that these $L_2$ analytic Fourier--Feynman transforms are actually linear operator isomorphisms from a Hilbert space into itself. We finally investigate the algebraic structures of these classes of the transforms on Wiener space, and show that they indeed are group isomorphic.

math.PR↗
arXiv2019

A space of generalized Brownian motion path-valued continuous functions with application

In this paper, we introduce the paths space $\mathcal C_0^{\mathrm{gBm}}$ which is consists of generalized Brownian motion path-valued continuous functions on $[0,T]$. We next present several relevant examples of the paths space integral. We then discuss the concept of the analytic Feynman integration theory for functionals $F$ on the paths space $\mathcal C_0^{\mathrm{gBm}}$.

math.FA↗
arXiv2019

Parts formulas involving the Fourier-Feynman transform associated with Gaussian process on Wiener space

In this paper, using a very general Cameron--Storvick theorem on the Wiener space $C_0[0,T]$, we establish various integration by parts formulas involving generalized analytic Feynman integrals, generalized analytic Fourier--Feynman transforms, and the first variation (associated with Gaussian processes) of functionals $F$ on $C_0[0,T]$ having the form $F(x)=f(\langle{α_1,x}\rangle, \ldots, \langle{α_n,x}\rangle)$ for scale almost every $x\in C_0[0,T]$, where $\langle{α,x}\rangle$ denotes the Paley--Wiener--Zygmund stochastic integral $\int_0^T α(t)dx(t)$, and $\{α_1,\ldots,α_n\}$ is an orthogonal set of nonzero functions in $L_2[0,T]$. The Gaussian processes used in this paper are not stationary.

math.FA↗
arXiv2017

Wiener integrals with respect to Yeh processes

We define Wiener integrals with respect to Yeh processes and study their properties. In particular, we obtain the martingale property of the associated stochastic processes and give a series expansion of Wiener integrals with respect to centered Yeh process. Moreover, we derive a representation of an Yeh process in terms of a random series.

math.PR↗
arXiv2015

Analytic Fourier--Feynman transforms and convolution type operations associated with Gaussian processes on Wiener space

In this paper we introduce the concept of a convolution type operation of functionals on Wiener space. It contains several kinds of the concepts of convolution products on Wiener space, which have been studied by many authors. We then investigate fundamental relationships between generalized analytic Fourier--Feynman transforms and convolution type operations. Both of the generalized analytic Fourier--Feynman transform of the convolution type operation and the convolution type operation of the generalized analytic Fourier--Feynman transforms are represented as a product of the generalized analytic Fourier--Feynman transforms.

math.PR↗