Analytic operator-valued generalized Feynman integral on function space

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 \[ dν_{δ,a}=\exp\{δ\mathrm{Var}(a)u^2\} du, \] 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 $C_{a,b}^{\prime }[0,T]$ are elements of the linear space \[ \bigcap_{δ>0} \mathcal L( L^1(\mathbb R,ν_{δ,a}),L^{\infty}(\mathbb R)). \]

arXiv comment24pages