A path-integral formulation is given of the Parisi-Wu method of stochastic quantization. The connection between the Langevin and Fokker-Planck equations is rederived in functional form bringing to light new aspects of these equations. The relation between this functional method and the recent approach by De-Alfaro-Fubini-Furlan is clarified. In particular we present the non-local transformation that connects the two formulations. In this dissertation we also investigate the hidden supersymmetry recently discovered by Parisi and Sourlas in Langevin processes. The Kelvin relations, for diffusion processes out of equilibrium, are shown to be a necessary and sufficient condition to have supersymmetry. This proves that this symmetry is a manifestation, at the path-integral level, of the Onsager principle of microreversibility present in the stochastic process. We analyze, in particular, the interesting interplay between forward and backward Fokker-Planck dynamics and, as an application, we rederive the fluctuation-dissipation theorem as a Ward-identity of this supersymmetry.
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