On the functional integral for generalized Wiener processes and nonequilibrium phenomena

The attention will be focussed on a generalized Wiener diffusion process for which the macroscopic evolution $\text{y}\text{?}\text{= c}{}_1\text{(}\text{y}\text{)}$ equals zero, of course, and where the variance of the process obeys $\text{g}\text{?}\text{s}{}^2\text{= c}{}_2\text{(}\text{y}\text{)}$. The diffusion function c₂(y) may be state dependent in an arbitrary way. We invoke our treatment of the general time-local Gaussian process as presented in a previous paper. This process will be seen to define a generalized functional Wiener measure. This measure has already been used implicitly in earlier work being concerned with nonlinear, nonequilibrium Markov processes. The sum of the generalized measure over the entire function space will be shown to be exactly related to the general Fokker-Planck equation for the driftless diffusion process. The relation between the well-defined functional sum and its corresponding functional integral will be studied in detail. The analysis demonstrates in clear fashion the origin of the deviations from other approaches, and provides an extension of our previous results on nonequilibrium, nonlinear phenomena to include generalized diffusion processes.