In the present paper we introduce positive flows and processes, which generalize ordinary dynamical systems and stochastic processes. The theory of positive flows and processes is based on the methods of positive operator theory and operator algebras theory. Its basic concepts are the phase and positive algebras, the spectral potential, dual entropy, equilibrium measures, action functional, sensitive states, and empirical measures. Within the frames of the theory we prove the law of large numbers with respect to the sensitive states and calculate asymptotics for the probabilities of large deviations in terms of the action functional.