Stochastic processes new foundations and representations in Hilbert space

In this thesis new foundations for the stochastic process are formulated which lead to the conventional stochastic formalism and in addition clearly define the notion of time reversibility for the stochastic process. The stochastic equations are shown to have representations in a separable Hilbert space H, which have no counterpart in unitary quantum dynamics. Beginning with an intuitive consideration of sequences of measurements, we define a time-ordered event space representing the collection of all imaginable outcomes for the measurement sequence. We then postulate the generalized distributive relation on the event space and examine the class of measurements for which this relation can be experimentally validated. The generalized distributive relation is shown to lead to a a-additive conditional probability on the event space and to a predictive and retrodictive formalism for stochastic processes. We show that the dynamics of the stochastic formalism are distinct from unitary quantum dynamics in several major ways. We propose the basis for a mathematical structure in H which would include both the stochastic formalism and the quantum formalism as special cases.