Analytic operator-valued generalized Feynman integral on function space

TL;DR

This paper develops an analytic operator-valued generalized Feynman integral on a broad class of Wiener spaces induced by generalized Brownian motion, establishing its existence and functional properties as an operator between specific function spaces.

Contribution

It introduces a new form of the generalized Feynman integral on complex Wiener spaces and investigates its existence and operator characteristics in a general setting.

Findings

Existence of the analytic operator-valued generalized Feynman integral as an operator from L^1 to L^∞.

The integral is well-defined for functionals involving stochastic Fourier--Stieltjes transforms.

The integral belongs to the intersection of operator spaces over all positive delta values.

Abstract

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 $L^1(\mathbb R, \nu_{\delta,a})$ to $L^{\infty}(\mathbb R)$ where $\nu_{\delta,a}$ is a $\sigma$-finite measure on $\mathbb R$ given by \[ d\nu_{\delta,a}=\exp\{\delta \mathrm{Var}(a)u^2\} du, \] where $\delta>0$ 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 \[ \bigcap_{\delta>0} \mathcal L( L^1(\mathbb R,\nu_{\delta,a}),L^{\infty}(\mathbb R)). \]