An Analogue of Wiener Measure and Its Applications

Abstract. Using a simple formula for conditional expectations over an analogue of Wiener space, we calculate a generalized analytic conditional Fourier-Feynman transform and convolution product of generalized cylinder functions which play important roles in Feynman integration theories and quantum mechanics. We then investigate their relationships, that is, the conditional Fourier-Feynman transform of the convolution product can be expressed in terms of the product of the conditional FourierFeynman transforms of each function. Finally we establish change of scale formulas for the generalized analytic conditional Fourier-Feynman transform and the conditional convolution product. In this evaluation formulas and change of scale formulas we use multivariate normal distributions so that the orthonormalization process of projection vectors which are essential to establish the conditional expectations, can be removed in the existing conditional Fourier-Feynman transforms, conditional convolution products and change of scale formulas.

Section: Introductionmentioning

confidence: 99%

“…This measure w ϕ is called an analogue of Wiener measure associated with the probability measure ϕ [9]. Let C and C + denote the sets of complex numbers and complex numbers with positive real parts, respectively.…”

Section: Introductionmentioning

confidence: 99%

A Conditional Fourier-Feynman Transform and Conditional Convolution Product With Change of Scales on a Function Space I

Cho¹

2017

“…Furthermore the author and his coauthors [6,8,11] introduced various kinds of the change of scale formulas for the conditional Wiener integrals of the function of the form F 1 defined on C 0 [0, t], C 0 (B), the infinite dimensional Wiener space and C[0, t], an analogue of Wiener space [9] which is the space of real-valued continuous paths on [0, t].…”

Section: Introductionmentioning

confidence: 99%

Change of Scale Formulas for a Generalized Conditional Wiener Integral

Cho¹,

Yoo²

2016

Abstract. Let C[0, t] denote the space of real-valued continuous functions on [0, t] and define a random vector Zn :with h = 0 a.e. Using a simple formula for a conditional expectation on C[0, t] with Zn, we evaluate a generalized analytic conditional Wiener integral of the functionvr(s)dx(s)) for F in a Banach algebra and for Ψ = f + φ which need not be bounded or continuous, whereand φ is the Fourier transform of a measure of bounded variation over R r . Finally we establish various change of scale transformations for the generalized analytic conditional Wiener integrals of Gr with the conditioning function Zn.

“…For a positive real t let C = C[0, t] be the space of all real-valued continuous functions on the closed interval [0, t] with the supremum norm. Let (C[0, t], B(C[0, t]), w ϕ ) denote the analogue of Wiener space associated with the probability measure ϕ [9,11], where ϕ is a probability measure on B(R). …”

Section: Operator-valued Function Space Integrals 905mentioning

confidence: 99%

“…Further work extending the above L(L 1 , L ∞ )-theory with the conditional analytic Feynman integrals was studied by the author [5] over the space (C r [0, t], w r ϕ ) [9,11] of the continuous R r -valued paths on [0, t] which generalizes the space C r 0 [0, t]. In fact the author [4] introduced the conditional Wiener integral over C r [0, t] and derived a simple formula for the conditional Wiener integral with the conditioning function X n : C r [0, t] → R (n+1)r given by X n (x) = (x(t 0 ), x(t 1 ), .…”

Section: Introductionmentioning

confidence: 99%

Operator-Valued Function Space Integrals via Conditional Integrals on an Analogue Wiener Space Ii

Cho¹

2016

Abstract. In the present paper, using a simple formula for the conditional expectations given a generalized conditioning function over an analogue of vector-valued Wiener space, we prove that the analytic operatorvalued Feynman integrals of certain classes of functions over the space can be expressed by the conditional analytic Feynman integrals of the functions. We then provide the conditional analytic Feynman integrals of several functions which are the kernels of the analytic operator-valued Feynman integrals.