Functional-integral approach to Parisi-Wu stochastic quantization: Scalar theory

The 5th-time stochastic-quantization approach to field theory, recently proposed by Parisi and Wu, is put in a path-integral form. The procedure of taking the limit $\ensuremath{\tau}\ensuremath{\rightarrow}\ensuremath{\infty}$ is analyzed and based on new grounds through the introduction of the vacuum-vacuum generating functional. Different aspects of the interplay between forward and backward Fokker-Planck dynamics are studied in detail in connection with the supersymmetry recently discovered in Gaussian stochastic processes.