A Stochastic Approach to the Definition of the Path Integral Measure

Abstract: We attempt to define a Stochastic Path Integral in Lorentzian time by restricting the relevant domain of integration on $C([0,1],M)$ over a Riemannian configuration manifold (M,g). Through fibration, we define $L^2$-isometric flux spaces in which we consider a stochastic process over which we define a Gaussian measure with respect to which the Path Integral may be rigorously formulated as a functional integral. We prove equivalence to the Euclidean Path Integral theory and the Feynman-Kac theorem.