A Stochastic Approach to the Definition of the Path Integral Measure

We to define a Path Integral in Lorentzian time by restricting the relevant domain of integration on $C([0,1],M)$ over a Riemannian configuration manifold $(M,g)$ and considering the dynamics of a particle evolving between to fixed endpoints with a referential non-degenerate classical trajectory, formulating a framework around a quadratic Lagrangian. Through fibration, we reduce the infinite-dimensional space under consideration to an $L^2$-isometric flux spaces in which we consider a stochastic process associated to a Gaussian measure. The Path Integral is subsequently defined as an expectation value with respect to the Gaussian measure, allowing us to rigorously formulate the former as a functional integral. We prove mathematical correspondence between the Stochastic Path Integral and the Euclidean Path Integral theory formulated rigorously under the Feynman-Kac theorem.