A functional Stieltjes measure and generalized diffusion processes

Abstract

The sum over paths for a generalized Wiener diffusion process has been presented in a previous paper on the basis of the concept of time-local gaussian processes. This well-defined functional sum leads to a functional integral for the probability of passing from an initial state to some final state of the system. A mathematical ambiguity remained in the freedom to choose the diffusion function in the measure anywhere between the so-called prepoint for each infinitesimal transition. In the present paper we will demonstrate, for the generalized diffusion process, how a logical extension of functional integration in the sense of Stieltjes quite naturally leads to an unambiguous formulation which has the special and interesting property of being form-invariant under nonlinear transformations according to the rules of ordinary differential calculus.